# LaTeX2HTML 2002-2-1 (1.71)
# Associate images original text with physical files.


$key = q/P({{bf{r},{{bf{p}),d^3{{bf{r},d^3{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1275.png"
 ALT="$P({\bf r}, {\bf p})  d^3{\bf r} d^3{\bf p}$">|; 

$key = q/{displaymath}p=frac{1}{beta}frac{partialln{mitOmega}}{partialV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img725.png"
 ALT="\begin{displaymath}
p = \frac{1}{\beta} \frac{\partial \ln {\mit\Omega}}{\partial V}.
\end{displaymath}">|; 

$key = q/10^{17};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1136.png"
 ALT="$10^{17}$">|; 

$key = q/{displaymath}exp(beta,hbar,omega)gg1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1105.png"
 ALT="\begin{displaymath}
\exp(\beta \hbar \omega) \gg 1,
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{T}left(frac{partial^2E}{partialVpartialT}right)=left[-frac{(frac{partial^2E}{partialTpartialV}right)right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img748.png"
 ALT="\begin{displaymath}
\frac{1}{T} \left(\frac{\partial ^2 E}{\partial V \partial T...
...eft (\frac{\partial^2 E}{\partial T \partial V}\right)\right].
\end{displaymath}">|; 

$key = q/hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1161.png"
 ALT="$\hbar$">|; 

$key = q/{displaymath}V^{(0)}=V+V'={{rm{constant},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img662.png"
 ALT="\begin{displaymath}
V^{(0)} = V + V' = {\rm constant},
\end{displaymath}">|; 

$key = q/B_2<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$B_2&lt;0$">|; 

$key = q/deltaRequivR(E+deltaE)-R(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="$\delta R \equiv
R(E+\delta E)- R(E)$">|; 

$key = q/{displaymath}{mitDelta}bar{E}=-W,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="\begin{displaymath}
{\mit\Delta} \bar{E} = -W,
\end{displaymath}">|; 

$key = q/n-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1165.png"
 ALT="$n-1$">|; 

$key = q/{displaymath}alpha_ssimeqalpha{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1406.png"
 ALT="\begin{displaymath}
\alpha_s \simeq \alpha
\end{displaymath}">|; 

$key = q/{displaymath}S=nuRleft[lnV+frac{3}{2}lnT+sigmaright],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1040.png"
 ALT="\begin{displaymath}
S = \nu R \left[ \ln V + \frac{3}{2}\ln T + \sigma\right],
\end{displaymath}">|; 

$key = q/{displaymath}k,theta_D=hbar,omega_D.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1255.png"
 ALT="\begin{displaymath}
k \theta_D = \hbar  \omega_D.
\end{displaymath}">|; 

$key = q/10^{15};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1135.png"
 ALT="$10^{15}$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=dbar{E}+{mathchar'26mskip-12mud}W,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img587.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = d\bar{E} + {\mathchar'26\mskip-12mud}W,
\end{displaymath}">|; 

$key = q/{{bf{v}={{bf{p}slashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1281.png"
 ALT="${\bf v} = {\bf p}/m$">|; 

$key = q/{{bf{k}!cdot!{{bf{E}_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1466.png"
 ALT="${\bf k}\!\cdot\! {\bf E}_0 = 0$">|; 

$key = q/{displaymath}lnZ^{(0)}=lnZ+lnZ',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1016.png"
 ALT="\begin{displaymath}
\ln Z^{(0)} = \ln Z + \ln Z',
\end{displaymath}">|; 

$key = q/>10^{20};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1141.png"
 ALT="$&gt;10^{20}$">|; 

$key = q/{displaymath}beta(tilde{E})=beta'(tilde{E}'),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="\begin{displaymath}
\beta (\tilde{E}) = \beta'(\tilde{E}'),
\end{displaymath}">|; 

$key = q/,{mathchar'26mskip-12mud}G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="$ {\mathchar'26\mskip-12mud}G$">|; 

$key = q/{displaymath}chi(E)proptoR^{,3N-1}slashE^{,1slash2}proptoE^{,3Nslash2-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="\begin{displaymath}
\chi(E) \propto R^{ 3N-1}/E^{ 1/2} \propto E^{ 3N/2-1}.
\end{displaymath}">|; 

$key = q/f(u_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$f(u_1)$">|; 

$key = q/2,p,q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$2 p q$">|; 

$key = q/{displaymath}p=frac{N,k,T}{V}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img726.png"
 ALT="\begin{displaymath}
p = \frac{N k  T}{V}.
\end{displaymath}">|; 

$key = q/exp[-beta,(n_1,epsilon_1+n_2,epsilon_2+cdots)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1386.png"
 ALT="$\exp[-\beta (n_1 \epsilon_1+n_2 \epsilon_2+\cdots)]$">|; 

$key = q/{displaymath}Q+Q'=0:{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img516.png"
 ALT="\begin{displaymath}
Q+ Q' =0:
\end{displaymath}">|; 

$key = q/{displaymath}rho({{bf{k}),d^3{{bf{k}=frac{d^3{{bf{k}}{(2pi)^3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1470.png"
 ALT="\begin{displaymath}
\rho({\bf k}) d^3{\bf k} = \frac{d^3{\bf k}}{(2\pi)^3}.
\end{displaymath}">|; 

$key = q/displaystyledx_i,dy_i,dz_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="$\displaystyle dx_i dy_i dz_i,$">|; 

$key = q/{displaymath}overline{E}=frac{sum_rexp(-beta,E_r),E_r}{sum_rexp(-beta,E_r)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img976.png"
 ALT="\begin{displaymath}
\overline{E} = \frac{\sum_r \exp(-\beta  E_r)  E_r}{\sum_r \exp(-\beta  E_r)},
\end{displaymath}">|; 

$key = q/omega_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1153.png"
 ALT="$\omega_z$">|; 

$key = q/{displaymath}dV=sum_id{{bf{S}_i!cdot!d{{bf{x}_i{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="\begin{displaymath}
dV = \sum_i d{\bf S}_i\!\cdot\! d{\bf x}_i
\end{displaymath}">|; 

$key = q/{displaymath}overline{frac{1}{2},m,v_z^{~2}}=frac{1}{2},k,T,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1304.png"
 ALT="\begin{displaymath}
\overline{\frac{1}{2} m  v_z^{&nbsp;2}} = \frac{1}{2}  k T,
\end{displaymath}">|; 

$key = q/displaystylep,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$\displaystyle p,$">|; 

$key = q/E_rllE^{(0)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img925.png"
 ALT="$E_r\ll E^{(0)}$">|; 

$key = q/epsilon=mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1540.png"
 ALT="$\epsilon=\mu$">|; 

$key = q/textstyle<;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$\textstyle &lt;$">|; 

$key = q/{displaymath}lnZ=lnZ_{{rm{MB}-lnN!,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1458.png"
 ALT="\begin{displaymath}
\ln Z = \ln Z_{\rm MB} - \ln N!,
\end{displaymath}">|; 

$key = q/{displaymath}bar{n}_s=-frac{1}{beta}frac{partial}{partialepsilon_s}left(lnsum_{n_s}{{rm{e}^{-beta,n_s,epsilon_s}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1421.png"
 ALT="\begin{displaymath}
\bar{n}_s =-\frac{1}{\beta}\frac{\partial}{\partial \epsilon_s}\left(\ln\sum_{n_s}
{\rm e}^{-\beta n_s \epsilon_s}\right).
\end{displaymath}">|; 

$key = q/2times10^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1134.png"
 ALT="$2\times 10^4$">|; 

$key = q/3!=6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$3!=6$">|; 

$key = q/{displaymath}P(omega),domega=frac{hbar}{4pi^2,c^2}frac{omega^3,domega}{exp(beta,hbar,omega)-1}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1499.png"
 ALT="\begin{displaymath}
P(\omega) d\omega = \frac{\hbar}{4\pi^2  c^2} \frac{\omega^3 d\omega}
{\exp(\beta \hbar \omega)-1}
\end{displaymath}">|; 

$key = q/sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1494.png"
 ALT="$\sin\theta$">|; 

$key = q/{displaymath}{mitOmega}^{(0)}={mitOmega}(E),{mitOmega}'(E^{(0)}-E).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="\begin{displaymath}
{\mit\Omega}^{(0)} = {\mit\Omega}(E)  {\mit\Omega}'(E^{(0)} - E).
\end{displaymath}">|; 

$key = q/{displaymath}S=kln{mitOmega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img583.png"
 ALT="\begin{displaymath}
S = k\ln{\mit\Omega}.
\end{displaymath}">|; 

$key = q/h=VslashA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$h = V/A$">|; 

$key = q/displaystyleoverline{epsilon}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1072.png"
 ALT="$\displaystyle \overline{\epsilon}_i$">|; 

$key = q/Tggtheta_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1256.png"
 ALT="$T\gg \theta_D$">|; 

$key = q/bar{X}slashbar{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img553.png"
 ALT="$\bar{X}/\bar{E}$">|; 

$key = q/E^{(0)}-E-deltaE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="$E^{(0)}- E -\delta E$">|; 

$key = q/E^f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img535.png"
 ALT="$E^f$">|; 

$key = q/{displaymath}dF=F(x+dx,y+dy)-F(x,y),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="\begin{displaymath}
dF = F(x + dx, y+dy)- F(x,y),
\end{displaymath}">|; 

$key = q/{displaymath}ln{calP}=lnN!-lnn!-ln,(N-n)!+nlnp+(N-n)lnq.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="\begin{displaymath}
\ln {\cal P} = \ln N! - \ln n!- \ln  (N-n)! +n\ln p +(N-n)\ln q.
\end{displaymath}">|; 

$key = q/Z(N);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1407.png"
 ALT="$Z(N)$">|; 

$key = q/m,a^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1159.png"
 ALT="$m  a^2$">|; 

$key = q/{displaymath}psi(q_1,cdots,q_f,s_1,cdots,s_g,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="\begin{displaymath}
\psi(q_1,\cdots, q_f, s_1,\cdots, s_g, t),
\end{displaymath}">|; 

$key = q/{displaymath}beta,muequivfrac{mu}{k,T}gg1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1533.png"
 ALT="\begin{displaymath}
\beta \mu \equiv \frac{\mu}{k T} \gg 1.
\end{displaymath}">|; 

$key = q/{displaymath}c_V=frac{1}{nu,k,T^2}int_0^inftysigma_D(omega),hbar,omegaleft[frac{omega}{[exp(beta,hbar,omega)-1]^2}right],domega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1247.png"
 ALT="\begin{displaymath}
c_V = \frac{1}{\nu  k  T^2} \int_0^\infty \sigma_D (\omega...
...r \omega}
{[\exp(\beta \hbar \omega)-1]^2}\right] d\omega.
\end{displaymath}">|; 

$key = q/{displaymath}p,dV+V,dp=nu,R,dT.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img790.png"
 ALT="\begin{displaymath}
p dV + V dp = \nu  R  dT.
\end{displaymath}">|; 

$key = q/q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$q$">|; 

$key = q/{displaymath}frac{partialln{mitOmega}}{partialx_alpha}=beta,bar{X}_alpha,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img558.png"
 ALT="\begin{displaymath}
\frac{\partial \ln{\mit\Omega}}{\partial x_\alpha} = \beta  \bar{X}_\alpha,
\end{displaymath}">|; 

$key = q/B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="$B$">|; 

$key = q/T^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1508.png"
 ALT="$T^4$">|; 

$key = q/k+dk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1225.png"
 ALT="$k+dk$">|; 

$key = q/displaystyleleft(sqrt{frac{2,pi,m}{beta}}right)^3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1029.png"
 ALT="$\displaystyle \left(\sqrt{\frac{2 \pi m}{\beta}}\right)^3,$">|; 

$key = q/{displaymath}bar{n}_s=frac{1}{{{rm{e}^{,alpha+beta,epsilon_s}pm1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1442.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_s} \pm 1},
\end{displaymath}">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="$t$">|; 

$key = q/{displaymath}M_C=1.4,M_odot.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1635.png"
 ALT="\begin{displaymath}
M_C = 1.4 M_\odot.
\end{displaymath}">|; 

$key = q/prightarrowbar{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img710.png"
 ALT="$p\rightarrow \bar{p}$">|; 

$key = q/{displaymath}beta,hbar,omega=frac{hbar,omega}{k,T}ll1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1099.png"
 ALT="\begin{displaymath}
\beta \hbar \omega = \frac{\hbar  \omega}{k T} \ll 1,
\end{displaymath}">|; 

$key = q/C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="$C$">|; 

$key = q/epsilon_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1068.png"
 ALT="$\epsilon_i$">|; 

$key = q/A_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="$A_1$">|; 

$key = q/{{rm{CO}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img775.png"
 ALT="${\rm CO}_2$">|; 

$key = q/displaystyleg(v_z),dv_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1296.png"
 ALT="$\displaystyle g(v_z) dv_z$">|; 

$key = q/epsilon=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1530.png"
 ALT="$\epsilon=0$">|; 

$key = q/Z_{{rm{MB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1456.png"
 ALT="$Z_{\rm MB}$">|; 

$key = q/{displaymath}sigma(omega),domega=2,frac{omega^2}{2pi^2,c^3},domega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1473.png"
 ALT="\begin{displaymath}
\sigma (\omega) d\omega = 2  \frac{\omega^2}{2\pi^2  c^3} d\omega.
\end{displaymath}">|; 

$key = q/(1slash2),R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1117.png"
 ALT="$(1/2) R$">|; 

$key = q/R_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1613.png"
 ALT="$R_\ast$">|; 

$key = q/Tequiv(k,beta)^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1070.png"
 ALT="$T\equiv (k \beta)^{-1}$">|; 

$key = q/{displaymath}lnZ=Nlnzeta-NlnN+N,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1057.png"
 ALT="\begin{displaymath}
\ln Z = N \ln \zeta - N \ln N + N,
\end{displaymath}">|; 

$key = q/p^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$p^3$">|; 

$key = q/{displaymath}lnn!simeqn,lnn-n+{calO}(1),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="\begin{displaymath}
\ln n! \simeq n \ln n - n +{\cal O}(1),
\end{displaymath}">|; 

$key = q/5.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img831.png"
 ALT="$5.8$">|; 

$key = q/{mitOmega}_f={mitOmega}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img616.png"
 ALT="${\mit\Omega}_f = {\mit\Omega}_i$">|; 

$key = q/(p+q)^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$(p+q)^4$">|; 

$key = q/{displaymath}p=~_{lt,Nrightarrowinfty~}frac{n_1}{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="\begin{displaymath}
p=&nbsp;_{lt N\rightarrow\infty&nbsp;}\frac{n_1}{N}.
\end{displaymath}">|; 

$key = q/6!;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$6!$">|; 

$key = q/beta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="$\beta'$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=dE+{mathchar'26mskip-12mud}W,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img712.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = dE +{\mathchar'26\mskip-12mud}W,
\end{displaymath}">|; 

$key = q/m_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1607.png"
 ALT="$m_p$">|; 

$key = q/bar{p}=bar{p}(V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="$\bar{p} = \bar{p}(V)$">|; 

$key = q/d{{bf{S}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="$d{\bf S}_i$">|; 

$key = q/{displaymath}H=H_0+H_1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="\begin{displaymath}
H = H_0 + H_1,
\end{displaymath}">|; 

$key = q/gammarightarrow1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img855.png"
 ALT="$\gamma\rightarrow 1$">|; 

$key = q/Z';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1017.png"
 ALT="$Z'$">|; 

$key = q/n_1,n_2,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1385.png"
 ALT="$n_1, n_2,\cdots$">|; 

$key = q/dS^{(0)}geq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img659.png"
 ALT="$dS^{(0)} \geq 0$">|; 

$key = q/s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$s$">|; 

$key = q/omegaslashc_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1228.png"
 ALT="$\omega/c_s$">|; 

$key = q/{displaymath}S=nuRleft[lnV-frac{3}{2}lnbeta+frac{3}{2}ln!left(frac{2,pi,m}{h_0^{~2}}right)+frac{3}{2}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1039.png"
 ALT="\begin{displaymath}
S = \nu R \left[ \ln V -\frac{3}{2} \ln \beta + \frac{3}{2}\...
...eft(\frac{2 \pi  m }{h_0^{&nbsp;2}}\right) + \frac{3}{2} \right],
\end{displaymath}">|; 

$key = q/(3slash2),k,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1063.png"
 ALT="$(3/2) k T$">|; 

$key = q/{displaymath}zeta=Vleft(frac{2,pi,m}{h_0^{~2},beta}right)^{3slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1030.png"
 ALT="\begin{displaymath}
\zeta = V \left( \frac{2 \pi  m}{h_0^{&nbsp;2} \beta}\right)^{3/2},
\end{displaymath}">|; 

$key = q/{displaymath}mu_0=frac{p_F^{,2}}{2,m}=frac{hbar^2,k_F^{,2}}{2,m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1552.png"
 ALT="\begin{displaymath}
\mu_0 = \frac{p_F^{ 2}}{2 m} = \frac{\hbar^2 k_F^{ 2}}{2 m}.
\end{displaymath}">|; 

$key = q/overline{n_1}pm{mitDelta}^astn_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="$\overline{n_1}\pm {\mit\Delta}^\ast n_1$">|; 

$key = q/{displaymath}E+E'simeqE^{(0)}={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="\begin{displaymath}
E + E' \simeq E^{(0)} = {\rm constant}.
\end{displaymath}">|; 

$key = q/{displaymath}sum_{n_1=0}^Nfrac{N!}{n_1!,(N-n_1)!},p^{n_1},q^{N-n_1},n_1equivp,frfrac{N!}{n_1!,(N-n_1)!},p^{n_1},q^{N-n_1}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="560" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="\begin{displaymath}
\sum_{n_1=0}^N\frac{N!}{n_1! (N-n_1)!} p^{n_1} q^{N-n_1}\...
..._1=0}^N
\frac{N!}{n_1! (N-n_1)!} p^{n_1} q^{N-n_1}
\right].
\end{displaymath}">|; 

$key = q/n_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1206.png"
 ALT="$n_x$">|; 

$key = q/R_odotsimeq7times10^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1510.png"
 ALT="$R_\odot\simeq
7\times 10^5$">|; 

$key = q/Y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="$Y'$">|; 

$key = q/S_0=kln{mitOmega}_s=nu,R,ln2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img703.png"
 ALT="$S_0 = k\ln{\mit\Omega}_s =
\nu  R  \ln 2$">|; 

$key = q/{displaymath}chi(E)proptoint^{E+deltaE}_Ed^3{{bf{p_1}cdotsd^3{{bf{p_N}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="\begin{displaymath}
\chi(E) \propto \int^{E+\delta E}_E d^3 {\bf p_1}\cdots d^3 {\bf p_N}
\end{displaymath}">|; 

$key = q/{displaymath}bar{epsilon}llk,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="29" BORDER="0"
 SRC="|."$dir".q|img1486.png"
 ALT="\begin{displaymath}
\bar{\epsilon} \ll k T.
\end{displaymath}">|; 

$key = q/{displaymath}bar{n}_s=frac{sum_{n_s}n_s,{{rm{e}^{-beta,n_s,epsilon_s}}{sum_{n_s}{{rm{e}^{-beta,n_s,epsilon_s}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1420.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{\sum_{n_s} n_s {\rm e}^{-\beta n_s \epsilon_s}}
{\sum_{n_s} {\rm e}^{-\beta n_s \epsilon_s}}.
\end{displaymath}">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=dE+{mathchar'26mskip-12mud}W=dE+bar{p},dV,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img665.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = d E + {\mathchar'26\mskip-12mud}W = dE + \bar{p} dV,
\end{displaymath}">|; 

$key = q/deltaE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$\delta E$">|; 

$key = q/p^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$p^2$">|; 

$key = q/{displaymath}{mitPsi}(cdotsQ_icdotsQ_jcdots)={mitPsi}(cdotsQ_jcdotsQ_icdots).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1367.png"
 ALT="\begin{displaymath}
{\mit\Psi}(\cdots Q_i\cdots Q_j\cdots)=
{\mit\Psi}(\cdots Q_j\cdots Q_i\cdots).
\end{displaymath}">|; 

$key = q/lnZ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1058.png"
 ALT="$\ln Z$">|; 

$key = q/{displaymath}overline{p}=frac{1}{beta}frac{partiallnZ}{partialV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img998.png"
 ALT="\begin{displaymath}
\overline{p} = \frac{1}{\beta} \frac{\partial \ln Z}{\partial V}.
\end{displaymath}">|; 

$key = q/N!simeqN,lnN-N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1460.png"
 ALT="$N!\simeq N \ln N - N$">|; 

$key = q/{displaymath}left[frac{partialln{mitOmega}'}{partialE'}right]_0equivbeta{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img928.png"
 ALT="\begin{displaymath}
\left[\frac{\partial \ln {\mit\Omega}'}{\partial E'} \right]_0 \equiv \beta
\end{displaymath}">|; 

$key = q/displaystyleB_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="$\displaystyle B_2$">|; 

$key = q/{displaymath}alphasimeqfrac{partiallnZ}{partialN}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1408.png"
 ALT="\begin{displaymath}
\alpha\simeq \frac{\partial \ln Z}{\partial N}.
\end{displaymath}">|; 

$key = q/K;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1593.png"
 ALT="$K$">|; 

$key = q/sum^{(s)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1419.png"
 ALT="$\sum^{(s)}$">|; 

$key = q/(partialE_rslashpartialx),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img541.png"
 ALT="$(\partial E_r/\partial x)  dx$">|; 

$key = q/{mitOmega}_{{rm{tot}^{(0)}sim{mitOmega}^{(0)}(tilde{E});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img652.png"
 ALT="${\mit\Omega}_{\rm tot}^{(0)}\sim {\mit\Omega}^{(0)}(\tilde{E})$">|; 

$key = q/displaystylexi_{{rm{FD};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1378.png"
 ALT="$\displaystyle \xi_{\rm FD}$">|; 

$key = q/F_f-F_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="$F_f- F_i$">|; 

$key = q/epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1529.png"
 ALT="$\epsilon$">|; 

$key = q/{displaymath}c_V=frac{1}{nu}left(frac{partialoverline{E}}{partialT}right)_V=fract(frac{partialoverline{E}}{partialbeta}right)_V,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1188.png"
 ALT="\begin{displaymath}
c_V = \frac{1}{\nu}\left(\frac{\partial \overline{E}}{\parti...
...} \left(
\frac{\partial\overline{E}}{\partial \beta}\right)_V,
\end{displaymath}">|; 

$key = q/{displaymath}Esimeqfrac{A-B}{R}+C,R,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1625.png"
 ALT="\begin{displaymath}
E \simeq \frac{A-B}{R} + C R,
\end{displaymath}">|; 

$key = q/{displaymath}frac{{mathchar'26mskip-12mud}Q}{T}equivdS.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="\begin{displaymath}
\frac{{\mathchar'26\mskip-12mud}Q}{T} \equiv dS.
\end{displaymath}">|; 

$key = q/omega_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1151.png"
 ALT="$\omega_x$">|; 

$key = q/{{bf{p}_i^{~2}=p_{i,1}^{~~2}+p_{i,2}^{~~2}+p_{i,3}^{~~2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="${\bf p}_i^{&nbsp;2} = p_{i 1}^{&nbsp;&nbsp;2}+p_{i 2}^{&nbsp;&nbsp;2}+p_{i 3}^{&nbsp;&nbsp;2}$">|; 

$key = q/(1slash2),nu,R,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1113.png"
 ALT="$(1/2) \nu  R  T$">|; 

$key = q/{displaymath}etaequivfrac{w}{q_1}leq1-frac{T_2}{T_1}=frac{T_1-T_2}{T_1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img886.png"
 ALT="\begin{displaymath}
\eta \equiv \frac{w}{q_1} \leq 1 - \frac{T_2} {T_1} = \frac{T_1 - T_2}{T_1}.
\end{displaymath}">|; 

$key = q/2,V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1044.png"
 ALT="$2 V$">|; 

$key = q/{mitOmega}proptoV^N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1051.png"
 ALT="${\mit\Omega}\propto
V^N$">|; 

$key = q/omega_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1152.png"
 ALT="$\omega_y$">|; 

$key = q/{calP}(n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="${\cal P}(n)$">|; 

$key = q/p=nu,R,TslashV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img798.png"
 ALT="$p = \nu  R T/V$">|; 

$key = q/{{bf{F};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="${\bf F}$">|; 

$key = q/q_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img877.png"
 ALT="$q_1$">|; 

$key = q/gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img768.png"
 ALT="$\gamma$">|; 

$key = q/{displaymath}frac{1}{k,T}=frac{partialln{mitOmega}}{partialE}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img731.png"
 ALT="\begin{displaymath}
\frac{1}{k T} = \frac{\partial \ln {\mit\Omega}}{\partial E}.
\end{displaymath}">|; 

$key = q/{displaymath}mu_0=frac{hbar^2}{2,m}left(3,pi^2,frac{N}{V}right)^{2slash3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1566.png"
 ALT="\begin{displaymath}
\mu_0 = \frac{\hbar^2}{2 m}\left(3 \pi^2 \frac{N}{V}\right)^{2/3}.
\end{displaymath}">|; 

$key = q/J>1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img972.png"
 ALT="$J&gt;1/2$">|; 

$key = q/n_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1216.png"
 ALT="$n_z$">|; 

$key = q/{{bf{p}=m,{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1143.png"
 ALT="${\bf p} = m  {\bf v}$">|; 

$key = q/{displaymath}c_Vsim3,Rleft(frac{theta_E}{T}right)^2,exp(-theta_EslashT).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1195.png"
 ALT="\begin{displaymath}
c_V \sim 3  R \left(\frac{\theta_E}{T}\right)^2 \exp(-\theta_E / T).
\end{displaymath}">|; 

$key = q/{displaymath}g(v_z),dv_z=nleft(frac{m}{2pi,k,T}right)^{1slash2}exp(-m,v_z^{~2}slash2,k,T),dv_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1300.png"
 ALT="\begin{displaymath}
g(v_z) dv_z = n \left(\frac{m}{2\pi  k T} \right)^{1/2} \exp(-m  v_z^{&nbsp;2}/2 k T) 
dv_z.
\end{displaymath}">|; 

$key = q/{displaymath}overline{u}(omega),domega=frac{hbar}{pi^2,c^3}frac{omega^3,domega}{exp(beta,hbar,omega)-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1488.png"
 ALT="\begin{displaymath}
\overline{u}(\omega) d\omega = \frac{\hbar}{\pi^2  c^3} \frac{\omega^3 d\omega}
{\exp(\beta \hbar \omega) - 1}.
\end{displaymath}">|; 

$key = q/mu_0ggk,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1567.png"
 ALT="$\mu_0\gg k T$">|; 

$key = q/KproptoM^{4slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1619.png"
 ALT="$K\propto M^{4/3}$">|; 

$key = q/{displaymath}{calP}(overline{n_1})int_{-infty}^{infty}!exp!left[-frac{(n-overlina}^astn_1int_{-infty}^{infty}exp(-x^2),dxsimeq1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="580" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="\begin{displaymath}
{\cal P}(\overline{n_1})\int_{-\infty}^{\infty}\!\exp\!
\lef...
...Delta}^\ast n_1\int_{-\infty}^{\infty}
\exp(-x^2) dx\simeq 1.
\end{displaymath}">|; 

$key = q/{displaymath}deltaq_i,deltap_igeqh,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img688.png"
 ALT="\begin{displaymath}
\delta q_i  \delta p_i \geq h,
\end{displaymath}">|; 

$key = q/{displaymath}lnZ=-frac{1}{2},beta,hbar,omega-ln[1-exp(-beta,hbar,omega)]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1096.png"
 ALT="\begin{displaymath}
\ln Z = - \frac{1}{2} \beta \hbar \omega -\ln [1- \exp(-\beta \hbar \omega)]
\end{displaymath}">|; 

$key = q/{mitOmega}(X);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="${\mit\Omega}(X)$">|; 

$key = q/E=tilde{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="$E= \tilde{E}$">|; 

$key = q/3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img973.png"
 ALT="$3/2$">|; 

$key = q/k_x+dk_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1218.png"
 ALT="$k_x + d k_x$">|; 

$key = q/textstyleequiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$\textstyle \equiv$">|; 

$key = q/10^{11};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$10^{11}$">|; 

$key = q/displaystylefrac{bar{p}}{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img672.png"
 ALT="$\displaystyle \frac{\bar{p}}{T}$">|; 

$key = q/{displaymath}overline{n_1}=N,p.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="\begin{displaymath}
\overline{n_1} = N p.
\end{displaymath}">|; 

$key = q/{displaymath}c_V=c_V^{(e)}+c_V^{(L)}=gamma,T+A,T^3.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1588.png"
 ALT="\begin{displaymath}
c_V = c_V^{(e)} + c_V^{(L)} = \gamma T + A T^3.
\end{displaymath}">|; 

$key = q/E'=E^{(0)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img926.png"
 ALT="$E' = E^{(0)}$">|; 

$key = q/beta=(k,T)^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img929.png"
 ALT="$\beta = (k T)^{-1}$">|; 

$key = q/{displaymath}dS'=frac{dE'+bar{p}',dV'}{T'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img669.png"
 ALT="\begin{displaymath}
d S' = \frac{ dE' + \bar{p}' dV'}{T'}.
\end{displaymath}">|; 

$key = q/h_0rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="$h_0\rightarrow 0$">|; 

$key = q/{displaymath}c_V=frac{1}{nu}left(frac{partialE}{partialT}right)_V.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img759.png"
 ALT="\begin{displaymath}
c_V = \frac{1}{\nu}\left(\frac{\partial E}{\partial T}\right)_V.
\end{displaymath}">|; 

$key = q/{displaymath}int_{n_1-1slash2}^{n_1+1slash2}{calP}(n),dn=P_N(n_1),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="\begin{displaymath}
\int_{n_1-1/2}^{n_1+1/2} {\cal P}(n)  dn = P_N(n_1),
\end{displaymath}">|; 

$key = q/displaystyleleft(frac{partialS}{partialV}right)_T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img744.png"
 ALT="$\displaystyle \left(\frac{\partial S}{\partial V}\right)_T$">|; 

$key = q/{displaymath}Z=({{rm{e}^{-beta,epsilon_1}+{{rm{e}^{-beta,epsilon_2}+cdots)^N,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1439.png"
 ALT="\begin{displaymath}
Z = ({\rm e}^{-\beta \epsilon_1}+{\rm e}^{-\beta \epsilon_2}+
\cdots)^N,
\end{displaymath}">|; 

$key = q/(3slash2),R=12.47;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1172.png"
 ALT="$(3/2) R
= 12.47$">|; 

$key = q/{displaymath}dE=nu,c_V,dT{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img761.png"
 ALT="\begin{displaymath}
dE = \nu  c_V dT
\end{displaymath}">|; 

$key = q/R_odot=7times10^5,{{rm{km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1617.png"
 ALT="$R_\odot= 7\times 10^5 {\rm km}$">|; 

$key = q/h_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$h_1$">|; 

$key = q/{displaymath}frac{gamma}{gamma-1}frac{dT}{T}=-frac{mu,g}{R,T},dz,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img840.png"
 ALT="\begin{displaymath}
\frac{\gamma}{\gamma-1}\frac{dT}{T} = -\frac{\mu  g}{R T}  dz,
\end{displaymath}">|; 

$key = q/y_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img593.png"
 ALT="$y_\alpha$">|; 

$key = q/{displaymath}int_0^inftyfrac{eta^3,deta}{expeta-1}=frac{pi^4}{15}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1506.png"
 ALT="\begin{displaymath}
\int_0^\infty
\frac{\eta^3 d\eta}{\exp\eta -1} = \frac{\pi^4}{15}.
\end{displaymath}">|; 

$key = q/{displaymath}0leqbar{n}_sleq1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1416.png"
 ALT="\begin{displaymath}
0 \leq \bar{n}_s \leq 1,
\end{displaymath}">|; 

$key = q/y_1,cdots,y_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img591.png"
 ALT="$y_1, \cdots, y_n$">|; 

$key = q/n_r=0,1,2,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1418.png"
 ALT="$n_r=0,1,2,\cdots$">|; 

$key = q/P(y_k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="$P(y_k)$">|; 

$key = q/overline{n_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\overline{n_1}$">|; 

$key = q/{displaymath}X_{alpha,r}equiv-frac{partialE_r}{partialx_alpha}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="\begin{displaymath}
X_{\alpha r} \equiv -\frac{\partial E_r}{\partial x_\alpha}
\end{displaymath}">|; 

$key = q/omega_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1234.png"
 ALT="$\omega_D$">|; 

$key = q/{displaymath}dF=X(x,y),dx+Y(x,y),dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="\begin{displaymath}
dF = X(x,y)  dx + Y(x,y)  dy,
\end{displaymath}">|; 

$key = q/{displaymath}M_{OV}=4,M_C=6.9,M_odot.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1639.png"
 ALT="\begin{displaymath}
M_{OV} = 4 M_C = 6.9 M_\odot.
\end{displaymath}">|; 

$key = q/Q_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1342.png"
 ALT="$Q_i$">|; 

$key = q/{displaymath}C_{n_1}^N=frac{N!}{n_1!}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="\begin{displaymath}
C_{n_1}^N = \frac{N!}{n_1 !}.
\end{displaymath}">|; 

$key = q/{{bf{q}=(q_x,q_y,q_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="${\bf q}=(q_x,q_y,q_z)$">|; 

$key = q/v_z+dv_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1294.png"
 ALT="$v_z+dv_z$">|; 

$key = q/{displaymath}overline{c,f(u)}=c,overline{f(u)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="\begin{displaymath}
\overline{c  f(u)} = c \overline{f(u)}.
\end{displaymath}">|; 

$key = q/{displaymath}gammalnV+lnp={{rm{constant},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img796.png"
 ALT="\begin{displaymath}
\gamma \ln V + \ln p = {\rm constant},
\end{displaymath}">|; 

$key = q/A=pm1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1361.png"
 ALT="$A=\pm 1$">|; 

$key = q/sigma(E,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img542.png"
 ALT="$\sigma(E,x)$">|; 

$key = q/bar{n}_rll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1445.png"
 ALT="$\bar{n}_r\ll 1$">|; 

$key = q/Fproptoexp(-beta,epsilon);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1571.png"
 ALT="$F\propto \exp(-\beta \epsilon)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{pv.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="341" BORDER="0"
 SRC="|."$dir".q|img892.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{pv.eps}}
\end{figure}">|; 

$key = q/{displaymath}S^ast=S{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img641.png"
 ALT="\begin{displaymath}
S^\ast = S
\end{displaymath}">|; 

$key = q/S=k,ln{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1052.png"
 ALT="$S= k  \ln {\mit\Omega}$">|; 

$key = q/10^{20};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1139.png"
 ALT="$10^{20}$">|; 

$key = q/beta_fequivbeta(bar{E}_f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img509.png"
 ALT="$\beta_f\equiv \beta(\bar{E}_f)$">|; 

$key = q/{displaymath}dln{mitOmega}=frac{partialln{mitOmega}}{partialE},dbar{E}+sum_{alphtialln{mitOmega}}{partialx_alpha},dbar{x}_alpha.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img578.png"
 ALT="\begin{displaymath}
d \ln {\mit\Omega} = \frac{\partial \ln {\mit\Omega}}{\parti...
...partial \ln {\mit\Omega}}{\partial x_\alpha} d\bar{x}_\alpha.
\end{displaymath}">|; 

$key = q/0_{+};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img708.png"
 ALT="$0_{+}$">|; 

$key = q/{{bf{B};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img939.png"
 ALT="${\bf B}$">|; 

$key = q/c_V=3,R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1193.png"
 ALT="$c_V = 3 R$">|; 

$key = q/{displaymath}ln,[{mitOmega}(E),{mitOmega}'(E')]=ln,[{mitOmega}(tilde{E}),{mitOmeambda(tilde{E})+lambda'(tilde{E}')],eta^2+cdots.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="642" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="\begin{displaymath}
\ln [{\mit\Omega}(E) {\mit\Omega}'(E')] = \ln [{\mit\Omeg...
...2} 
[\lambda(\tilde{E})+\lambda'(\tilde{E}')] \eta^2+\cdots.
\end{displaymath}">|; 

$key = q/2.735~^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1501.png"
 ALT="$2.735&nbsp;^\circ$">|; 

$key = q/{displaymath}lnZ=alpha,N+sum_rln,(1+{{rm{e}^{-alpha-beta,epsilon_r}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1417.png"
 ALT="\begin{displaymath}
\ln Z = \alpha N + \sum_r\ln  (1+{\rm e}^{-\alpha-\beta \epsilon_r}),
\end{displaymath}">|; 

$key = q/mu_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1550.png"
 ALT="$\mu_0$">|; 

$key = q/P_N(n_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$P_N(n_1)$">|; 

$key = q/alpha+beta,epsilon_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1430.png"
 ALT="$\alpha+\beta \epsilon_s$">|; 

$key = q/{displaymath}Qequiv{mitDelta}bar{E}+W,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="\begin{displaymath}
Q \equiv {\mit\Delta} \bar{E} + W,
\end{displaymath}">|; 

$key = q/{displaymath}P(y_k)=frac{{mitOmega}(E;y_k)}{{mitOmega}(E)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="\begin{displaymath}
P(y_k) = \frac{{\mit\Omega}(E; y_k)}{{\mit\Omega}(E)}.
\end{displaymath}">|; 

$key = q/displaystylehbar,omega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1477.png"
 ALT="$\displaystyle \hbar \omega,$">|; 

$key = q/{mitOmega}(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img611.png"
 ALT="${\mit\Omega}(y)$">|; 

$key = q/q^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$q^3$">|; 

$key = q/{displaymath}omega_D=c_sleft(6pi^2,frac{N}{V}right)^{1slash3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1242.png"
 ALT="\begin{displaymath}
\omega_D = c_s\left(6 \pi^2  \frac{N}{V}\right)^{1/3}.
\end{displaymath}">|; 

$key = q/ngg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$n\gg 1$">|; 

$key = q/{displaymath}ln{mitOmega}(E)=ln{mitOmega}(tilde{E})+beta(tilde{E}),eta-frac{1}{2},lambda(tilde{E}),eta^2+cdots,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="\begin{displaymath}
\ln {\mit\Omega}(E) = \ln{\mit\Omega}(\tilde{E}) + \beta(\tilde{E}) \eta -
\frac{1}{2} \lambda
(\tilde{E}) \eta^2+\cdots,
\end{displaymath}">|; 

$key = q/{displaymath}overline{X}=-overline{frac{partialE_r}{partialx}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img994.png"
 ALT="\begin{displaymath}
\overline{X} = - \overline{\frac{\partial E_r}{\partial x}}
\end{displaymath}">|; 

$key = q/T_2<T_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img884.png"
 ALT="$T_2 &lt; T_1$">|; 

$key = q/4pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1516.png"
 ALT="$4\pi$">|; 

$key = q/h_0rightarrowh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img693.png"
 ALT="$h_0 \rightarrow h$">|; 

$key = q/partialE_rslashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img544.png"
 ALT="$\partial E_r/\partial x$">|; 

$key = q/(+);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img941.png"
 ALT="$(+)$">|; 

$key = q/c_VslashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1590.png"
 ALT="$c_V/T$">|; 

$key = q/displaystylefrac{1}{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img670.png"
 ALT="$\displaystyle \frac{1}{T}$">|; 

$key = q/{displaymath}p=p_0left(1-frac{gamma-1}{gamma}frac{z}{z_0}right)^{gammaslash(gamma-1)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img854.png"
 ALT="\begin{displaymath}
p= p_0 \left(1 - \frac{\gamma -1}{\gamma} \frac{z}{z_0} \right)^{\gamma/(\gamma-1)}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}({{mitSigma}}otimes{{mitSigma}})={mitOmega}({{mitSigma}}),{mitOmega}({{mitSigma}}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="\begin{displaymath}
{\mit\Omega}({{\mit\Sigma}}\otimes{{\mit\Sigma}}) = {\mit\Omega}({{\mit\Sigma}})  {\mit\Omega}({{\mit\Sigma}}).
\end{displaymath}">|; 

$key = q/E_r(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img540.png"
 ALT="$E_r(x)$">|; 

$key = q/{displaymath}H_0,{mitPsi}_r=E_r,{mitPsi}_r,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="\begin{displaymath}
H_0  {\mit\Psi}_r = E_r  {\mit\Psi}_r,
\end{displaymath}">|; 

$key = q/{displaymath}frac{dp}{p}=-frac{mu,g}{R,T},dz.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img813.png"
 ALT="\begin{displaymath}
\frac{dp}{p} = -\frac{\mu  g}{R T}  dz.
\end{displaymath}">|; 

$key = q/bar{p}(V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="$\bar{p}(V)$">|; 

$key = q/k_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1556.png"
 ALT="$k_F$">|; 

$key = q/beta_f'equivbeta'(bar{E}_f');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img510.png"
 ALT="$\beta_f'\equiv \beta'(\bar{E}_f')$">|; 

$key = q/E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="$E$">|; 

$key = q/{displaymath}P(E)=C,{mitOmega}(E),{mitOmega}'(E^{(0)}-E),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="\begin{displaymath}
P(E) = C  {\mit\Omega}(E)  {\mit\Omega}'(E^{(0)} - E),
\end{displaymath}">|; 

$key = q/displaystylesum_{n_1=0}^{N}frac{N!}{n_1!,(N-n_1)!},p^{n_1},q^{N-n_1},(n_1)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$\displaystyle \sum_{n_1=0}^{N}\frac{N!}{n_1! (N-n_1)!} p^{n_1} q^{N-n_1} (n_1)^2$">|; 

$key = q/{displaymath}R_ast=frac{(9pi)^{2slash3}}{8},frac{hbar^2}{m},frac{1}{G,m_p^{,5slash3},M^{1slash3}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img1615.png"
 ALT="\begin{displaymath}
R_\ast = \frac{(9\pi)^{2/3}}{8} \frac{\hbar^2}{m} \frac{1}{G m_p^{ 5/3} 
M^{1/3}}.
\end{displaymath}">|; 

$key = q/alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="$\alpha$">|; 

$key = q/dHslashdt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$d H/dt=0$">|; 

$key = q/q^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$q^2$">|; 

$key = q/V=L_xL_yL_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1223.png"
 ALT="$V = L_x L_y L_z$">|; 

$key = q/ygg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img952.png"
 ALT="$y\gg 1$">|; 

$key = q/S=kflnE+kflnV+cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img685.png"
 ALT="$S = kf\ln E + kf\ln V + \cdots$">|; 

$key = q/Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$Y$">|; 

$key = q/{displaymath}S'=2,S.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1048.png"
 ALT="\begin{displaymath}
S' = 2  S.
\end{displaymath}">|; 

$key = q/chi(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="$\chi(E)$">|; 

$key = q/g(v_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1307.png"
 ALT="$g(v_y)$">|; 

$key = q/{displaymath}n(omega,theta),domega,dtheta=frac{1}{2},n(omega),domega,sintheta,dtheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1493.png"
 ALT="\begin{displaymath}
n(\omega, \theta) d\omega d\theta =
\frac{1}{2} n(\omega) d\omega \sin\theta d\theta,
\end{displaymath}">|; 

$key = q/{mitOmega}'(E');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="${\mit\Omega}'(E')$">|; 

$key = q/{displaymath}yequivbeta,mu,B=frac{mu,B}{k,T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img949.png"
 ALT="\begin{displaymath}
y\equiv \beta \mu  B = \frac{\mu  B}{k T}.
\end{displaymath}">|; 

$key = q/dS={mathchar'26mskip-12mud}QslashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img667.png"
 ALT="$dS = {\mathchar'26\mskip-12mud}Q/T$">|; 

$key = q/{displaymath}bar{y}=frac{sum_k{mitOmega}(E;y_k),y_k}{{mitOmega}(E)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="\begin{displaymath}
\bar{y} = \frac{\sum_k {\mit\Omega}(E; y_k)  y_k}{{\mit\Omega}(E)},
\end{displaymath}">|; 

$key = q/ln,[{mitOmega}(E),{mitOmega}'(E')];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$\ln [{\mit\Omega}(E)  {\mit\Omega}'(E')]$">|; 

$key = q/{displaymath}int_0^inftyF(v),dv=n.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img1322.png"
 ALT="\begin{displaymath}
\int_0^\infty F(v) dv = n.
\end{displaymath}">|; 

$key = q/{displaymath}eta=frac{T_2}{T_1-T_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img915.png"
 ALT="\begin{displaymath}
\eta = \frac{T_2}{T_1 - T_2}.
\end{displaymath}">|; 

$key = q/{displaymath}overline{v}=4pileft(frac{m}{2pi,k,T}right)^{3slash2}left(frac{2,k,T}{m}right)^2int_0^inftyy^3exp(-y^2),dy.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1330.png"
 ALT="\begin{displaymath}
\overline{v} = 4\pi \left(\frac{m}{2\pi  k T}\right)^{3/2}...
...t(\frac{2 k T}{m}\right)^2 \int_0^\infty y^3 \exp(-y^2) dy.
\end{displaymath}">|; 

$key = q/{displaymath}ln{mitOmega}=lnB+NlnV+frac{3N}{2},lnE.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img778.png"
 ALT="\begin{displaymath}
\ln {\mit\Omega} = \ln B + N\ln V +\frac{3N}{2}  \ln E.
\end{displaymath}">|; 

$key = q/(p+q)^N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$(p+q)^N$">|; 

$key = q/{displaymath}Tsimfrac{E-E_0}{k,f},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img698.png"
 ALT="\begin{displaymath}
T \sim \frac{E - E_0}{k f},
\end{displaymath}">|; 

$key = q/{mitOmega}^{(0)}(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img643.png"
 ALT="${\mit\Omega}^{(0)}(E)$">|; 

$key = q/{displaymath}sum_{n=0}^inftyexp(-n,beta,hbar,omega)=frac{1}{1-exp(-beta,hbar,omega)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1094.png"
 ALT="\begin{displaymath}
\sum_{n=0}^\infty \exp(- n \beta  \hbar  \omega) = \frac{1}{1-\exp(-\beta \hbar
 \omega)}.
\end{displaymath}">|; 

$key = q/lambda<lambda_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1565.png"
 ALT="$\lambda&lt;\lambda_F$">|; 

$key = q/{displaymath}frac{dy}{dx}=-frac{X'}{Y'}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="\begin{displaymath}
\frac{dy}{dx} = -\frac{X'}{Y'}.
\end{displaymath}">|; 

$key = q/{displaymath}bar{n}_s=-frac{1}{beta}frac{partiallnZ}{partialepsilon_s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1388.png"
 ALT="\begin{displaymath}
\bar{n}_s = -\frac{1}{\beta}\frac{\partial \ln Z}{\partial\epsilon_s}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}(E,V)proptoV^Nchi(E).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img777.png"
 ALT="\begin{displaymath}
{\mit\Omega}(E, V) \propto V^N \chi(E).
\end{displaymath}">|; 

$key = q/6^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1523.png"
 ALT="$6^\circ$">|; 

$key = q/7slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img975.png"
 ALT="$7/2$">|; 

$key = q/{displaymath}dS=left.left(dbar{E}+sum_alphabar{X}_alpha,dbar{x}_alpharight)rightslashT,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img582.png"
 ALT="\begin{displaymath}
d S = \left.\left(d\bar{E} +\sum_\alpha \bar{X}_\alpha d\bar{x}_\alpha\right)\right/T,
\end{displaymath}">|; 

$key = q/{displaymath}S=nu,Rleft[lnfrac{V}{N}+frac{3}{2}lnT+sigma_0right]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1060.png"
 ALT="\begin{displaymath}
S = \nu  R \left[ \ln \frac{V}{N} + \frac{3}{2} \ln T + \sigma_0\right]
\end{displaymath}">|; 

$key = q/beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="$\beta$">|; 

$key = q/dT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img741.png"
 ALT="$dT$">|; 

$key = q/{displaymath}p=p_0expleft(-frac{z}{z_0}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img814.png"
 ALT="\begin{displaymath}
p = p_0 \exp\left(-\frac{z}{z_0}\right).
\end{displaymath}">|; 

$key = q/{displaymath}gamma,frac{dV}{V}+frac{dp}{p}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img794.png"
 ALT="\begin{displaymath}
\gamma \frac{dV}{V} + \frac{dp}{p}=0,
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{2},k,T,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1086.png"
 ALT="$\displaystyle \frac{1}{2}  k T,$">|; 

$key = q/{displaymath}lnZ=-sum_rln,(1-{{rm{e}^{-beta,epsilon_r}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1425.png"
 ALT="\begin{displaymath}
\ln Z = -\sum_r \ln (1-{\rm e}^{-\beta \epsilon_r}),
\end{displaymath}">|; 

$key = q/{displaymath}beta=frac{partialln{mitOmega}}{partialE},~~~~beta,bar{X}_alpha=frac{partialln{mitOmega}}{partialx_alpha}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img579.png"
 ALT="\begin{displaymath}
\beta = \frac{\partial \ln {\mit\Omega}}{\partial E},&nbsp;&nbsp;&nbsp;&nbsp;\be...
...{X}_\alpha
=\frac{\partial\ln {\mit\Omega}}{\partial x_\alpha}
\end{displaymath}">|; 

$key = q/{displaymath}gammaequivfrac{c_p}{c_V}=1+frac{R}{c_V}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img769.png"
 ALT="\begin{displaymath}
\gamma \equiv \frac{c_p}{c_V} = 1 +\frac{R}{c_V}
\end{displaymath}">|; 

$key = q/0.35;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img834.png"
 ALT="$0.35$">|; 

$key = q/{displaymath}c_s=sqrt{frac{gamma,p}{rho}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img770.png"
 ALT="\begin{displaymath}
c_s = \sqrt{\frac{\gamma  p}{\rho}},
\end{displaymath}">|; 

$key = q/{displaymath}c_V=frac{1}{nu}left(frac{partialE}{partialT}right)_V=frac{3}{2}R.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img784.png"
 ALT="\begin{displaymath}
c_V = \frac{1}{\nu} \left(\frac{\partial E}{\partial T}\right)_V = \frac{3}{2} R.
\end{displaymath}">|; 

$key = q/W_{sr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$W_{sr}$">|; 

$key = q/{displaymath}P(omega),domega=int_0^{pislash2}c,costheta,,hbar,omega,,n(omega,theta),domega,dtheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1496.png"
 ALT="\begin{displaymath}
P(\omega)  d\omega =\int_0^{\pi/2} c \cos\theta  \hbar \omega  n(\omega, \theta)
 d\omega d\theta,
\end{displaymath}">|; 

$key = q/5times10^6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1138.png"
 ALT="$5\times 10^6$">|; 

$key = q/bar{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="$\bar{E}$">|; 

$key = q/{displaymath}N_{{rm{eff}simeqfrac{k,T}{mu},N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1580.png"
 ALT="\begin{displaymath}
N_{\rm eff} \simeq \frac{k T}{\mu} N.
\end{displaymath}">|; 

$key = q/0.05;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1129.png"
 ALT="$0.05$">|; 

$key = q/{displaymath}frac{3V}{2pi^2,c_s^{~3}}int_0^{omega_D}omega^2,domega=frac{V}{2pi^2,c_s^{~3}},omega_D^{~3}=3,N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1241.png"
 ALT="\begin{displaymath}
\frac{3 V}{2\pi^2  c_s^{&nbsp;3}}\int_0^{\omega_D} \omega^2  d\omega = \frac{V}
{2\pi^2  c_s^{&nbsp;3}} \omega_D^{&nbsp;3} = 3 N.
\end{displaymath}">|; 

$key = q/{displaymath}E=frac{1}{2,m}sum_{i=1}^N{{bf{p}_i^{~2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="\begin{displaymath}
E = \frac{1}{2 m}\sum_{i=1}^N {\bf p}_i^{&nbsp;2},
\end{displaymath}">|; 

$key = q/{displaymath}{mitPsi}(cdotsQ_icdotsQ_jcdots)=A,{mitPsi}(cdotsQ_jcdotsQ_icdots),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1358.png"
 ALT="\begin{displaymath}
{\mit\Psi}(\cdots Q_i\cdots Q_j\cdots)=A {\mit\Psi}(\cdots Q_j\cdots Q_i\cdots),
\end{displaymath}">|; 

$key = q/y+deltay;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img609.png"
 ALT="$y+\delta y$">|; 

$key = q/{figure}{epsfysize{=3.5in{setbox{0=hbox{{epsffile{heat.eps}{}{centerline{rotu0}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="488" HEIGHT="406" BORDER="0"
 SRC="|."$dir".q|img1266.png"
 ALT="\begin{figure}
\epsfysize =3.5in\setbox 0=\hbox{\epsffile{heat.eps}}
\centerline{\rotu0}
\end{figure}">|; 

$key = q/{displaymath}exp(beta,hbar,omega)simeq1+beta,hbar,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1101.png"
 ALT="\begin{displaymath}
\exp(\beta \hbar \omega) \simeq 1 + \beta  \hbar  \omega,
\end{displaymath}">|; 

$key = q/{displaymath}B_k=left.frac{d^kln{calP}}{dn^k}right|_{n=tilde{n}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="\begin{displaymath}
B_k = \left.\frac{d^k \ln {\cal P}}{d n^k}\right\vert _{n=\tilde{n}}.
\end{displaymath}">|; 

$key = q/N_{{rm{eff};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1576.png"
 ALT="$N_{\rm eff}$">|; 

$key = q/bar{x}_alpha+dbar{x}_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img575.png"
 ALT="$\bar{x}_\alpha + d\bar{x}_\alpha$">|; 

$key = q/T'>T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img660.png"
 ALT="$T'&gt;T$">|; 

$key = q/{displaymath}frac{1}{k,T}=frac{partiallnchi}{partialE}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img732.png"
 ALT="\begin{displaymath}
\frac{1}{k T} = \frac{\partial \ln \chi}{\partial E}.
\end{displaymath}">|; 

$key = q/{displaymath}E=sum_{i=1}^Nfrac{p_i^{~2}}{2,m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1020.png"
 ALT="\begin{displaymath}
E = \sum_{i=1}^N \frac{ p_i^{&nbsp;2}}{2 m},
\end{displaymath}">|; 

$key = q/lambda';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="$\lambda'$">|; 

$key = q/{displaymath}bar{E}=tilde{E}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="\begin{displaymath}
\bar{E} = \tilde{E}.
\end{displaymath}">|; 

$key = q/{displaymath}epsilon=frac{p^2}{2,m}=frac{hbar^2,k^2}{2,m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1549.png"
 ALT="\begin{displaymath}
\epsilon = \frac{p^2}{2 m} =\frac{\hbar^2 k^2}{2 m},
\end{displaymath}">|; 

$key = q/h,nu={mitDelta}E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1119.png"
 ALT="$h  \nu = {\mit\Delta} E$">|; 

$key = q/{displaymath}bar{n}_s={{rm{e}^{-alpha-beta,epsilon_s}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1451.png"
 ALT="\begin{displaymath}
\bar{n}_s = {\rm e}^{-\alpha-\beta \epsilon_s}
\end{displaymath}">|; 

$key = q/{displaymath}P(bar{E}_f)>P(E_i),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img517.png"
 ALT="\begin{displaymath}
P(\bar{E}_f) &gt; P(E_i),
\end{displaymath}">|; 

$key = q/{displaymath}R=8.3143~~~{{rm{joulesslashkelvinslashmole}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img625.png"
 ALT="\begin{displaymath}
R= 8.3143&nbsp;&nbsp;&nbsp;{\rm joules/kelvin/mole}
\end{displaymath}">|; 

$key = q/{mitOmega}_{{rm{spin}(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img536.png"
 ALT="${\mit\Omega}_{\rm spin}(E)$">|; 

$key = q/{{rm{C}_2{{rm{H}_6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img776.png"
 ALT="${\rm C}_2 {\rm H}_6$">|; 

$key = q/1!=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$1! = 1$">|; 

$key = q/c_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$c_p$">|; 

$key = q/{displaymath}frac{partiallnP}{partialE}=frac{partialln{mitOmega}}{partialE}-beta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img937.png"
 ALT="\begin{displaymath}
\frac{\partial \ln P}{\partial E} = \frac{\partial \ln {\mit\Omega}}{\partial E}
-\beta =0,
\end{displaymath}">|; 

$key = q/{displaymath}overline{E^2}=frac{sum_rexp(-beta,E_r),E_r^{~2}}{sum_rexp(-beta,E_r)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img982.png"
 ALT="\begin{displaymath}
\overline{E^2} = \frac{\sum_r \exp(-\beta  E_r)  E_r^{&nbsp;2}}{\sum_r \exp(-\beta  E_r)}.
\end{displaymath}">|; 

$key = q/V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$V$">|; 

$key = q/{displaymath}Z=frac{V^N}{h_0^{~3N}}intcdotsintexp[-(betaslash2m),p_1^{~2}],d^3{{1cdotsexp[-(betaslash2m),p_N^{~2}],d^3{{bf{p}_N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1023.png"
 ALT="\begin{displaymath}
Z = \frac{V^N}{h_0^{&nbsp;3N}}\int\cdots \int \exp[-(\beta/2m) p...
...d^3{\bf p}_1
\cdots \exp[-(\beta/2m) p_N^{&nbsp;2}] d^3{\bf p}_N.
\end{displaymath}">|; 

$key = q/{displaymath}k_x,(x+L_x)=k_x,x+2,pi,n_x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1205.png"
 ALT="\begin{displaymath}
k_x  (x+ L_x) = k_x  x + 2 \pi  n_x,
\end{displaymath}">|; 

$key = q/{displaymath}bar{epsilon}simeqk,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1484.png"
 ALT="\begin{displaymath}
\bar{\epsilon} \simeq k T.
\end{displaymath}">|; 

$key = q/{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1197.png"
 ALT="${\bf r}$">|; 

$key = q/{mitOmega}(E,V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="${\mit\Omega}(E, V)$">|; 

$key = q/p(V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img891.png"
 ALT="$p(V)$">|; 

$key = q/M_odot=2times10^{30},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1618.png"
 ALT="$M_\odot = 2\times 10^{30} {\rm kg}$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W=sum_{alpha=1}^nbar{X}_alpha,dx_alpha,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W = \sum_{\alpha =1}^n \bar{X}_\alpha  dx_\alpha,
\end{displaymath}">|; 

$key = q/5times10^{14};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1133.png"
 ALT="$5 \times 10^{14}$">|; 

$key = q/{displaymath}frac{dlnn!}{dn}simeqfrac{ln,(n+1)!-lnn!}{1}=ln!left[frac{(n+1)!}{n!}right]=ln,(n+1),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="\begin{displaymath}
\frac{d\ln n!}{dn} \simeq \frac{\ln (n+1)!-\ln n!}{1} =
\ln\!\left[\frac{(n+1)!}{n!}\right] = \ln (n+1),
\end{displaymath}">|; 

$key = q/Nsim10^{24};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img605.png"
 ALT="$N\sim 10^{24}$">|; 

$key = q/bar{x}_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img573.png"
 ALT="$\bar{x}_\alpha$">|; 

$key = q/R=N_A,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1034.png"
 ALT="$R= N_A  k$">|; 

$key = q/textstyle0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1238.png"
 ALT="$\textstyle 0$">|; 

$key = q/{displaymath}f({{bf{r},{{bf{v}),d^3{{bf{r},d^3{{bf{v}=Cexp(-beta,m,v^2slash2),d^3{{bf{r},d^3{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1282.png"
 ALT="\begin{displaymath}
f({\bf r}, {\bf v}) d^3{\bf r} d^3{\bf v} = C \exp(-\beta   m  v^2/2) 
d^3{\bf r} d^3{\bf v},
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}(E,V)proptoV^fE^f.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="\begin{displaymath}
{\mit\Omega}(E, V) \propto V^f E^f.
\end{displaymath}">|; 

$key = q/Z=Z(beta,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1000.png"
 ALT="$Z= Z(\beta, x)$">|; 

$key = q/{displaymath}(p+q)^Nequivsum_{n=0}^{N}frac{N!}{n!,(N-n)!},p^n,q^{N-n}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="\begin{displaymath}
(p+q)^N \equiv \sum_{n=0}^{N} \frac{N!}{n! (N-n)!} p^n  q^{N-n}.
\end{displaymath}">|; 

$key = q/Trightarrowoverline{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img711.png"
 ALT="$T\rightarrow\overline{T}$">|; 

$key = q/{displaymath}overline{(n_1)^2}=sum_{n_1=0}^{N}frac{N!}{n_1!,(N-n_1)!},p^{n_1},q^{N-n_1},(n_1)^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="\begin{displaymath}
\overline{(n_1)^2}=\sum_{n_1=0}^{N}\frac{N!}{n_1! (N-n_1)!} p^{n_1} 
q^{N-n_1} (n_1)^2.
\end{displaymath}">|; 

$key = q/E^{(0)}+deltaE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img918.png"
 ALT="$E^{(0)} + \delta E$">|; 

$key = q/n_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1390.png"
 ALT="$n_s$">|; 

$key = q/exp,(alpha+beta,epsilon_r)gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1446.png"
 ALT="$\exp (\alpha +
\beta \epsilon_r)\gg 1$">|; 

$key = q/|v|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1317.png"
 ALT="$\vert v\vert$">|; 

$key = q/{mitOmega}simE^f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img637.png"
 ALT="${\mit\Omega} \sim E^f$">|; 

$key = q/{displaymath}frac{partialX'}{partialy}neqfrac{partialY'}{partialx}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="\begin{displaymath}
\frac{\partial X'}{\partial y} \neq \frac{\partial Y'}{\partial x}
\end{displaymath}">|; 

$key = q/{displaymath}ln{calP}(tilde{n}+eta)simeqln{calP}(tilde{n})+eta,B_1+frac{eta^2}{2},B_2+cdots,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="\begin{displaymath}
\ln {\cal P}(\tilde{n}+\eta) \simeq \ln{\cal P}(\tilde{n}) +\eta 
B_1+\frac{\eta^2}{2} B_2+\cdots ,
\end{displaymath}">|; 

$key = q/r_oplussimeq1.5times10^8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1514.png"
 ALT="$r_\oplus\simeq 1.5\times 10^8$">|; 

$key = q/S_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img706.png"
 ALT="$S_0$">|; 

$key = q/deltaq_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img689.png"
 ALT="$\delta q_i$">|; 

$key = q/v_{rmrms};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1326.png"
 ALT="$v_{\rm rms}$">|; 

$key = q/{displaymath}{{rm{Average~Age}simeqP_{18}times18+P_{19}times19+P_{20}times20+cdots.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="\begin{displaymath}
{\rm Average&nbsp;Age} \simeq P_{18}\times 18 + P_{19}\times 19 + P_{20}\times 20
+\cdots.
\end{displaymath}">|; 

$key = q/-qslashT_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img873.png"
 ALT="$-q/T_1$">|; 

$key = q/betarightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1448.png"
 ALT="$\beta\rightarrow 0$">|; 

$key = q/{displaymath}M_{OV}=mbox{1.5---2.5},M_odot.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1640.png"
 ALT="\begin{displaymath}
M_{OV} = \mbox{1.5--2.5} M_\odot.
\end{displaymath}">|; 

$key = q/BslashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$B/T$">|; 

$key = q/displaystyles_j.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1352.png"
 ALT="$\displaystyle s_j.$">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$p$">|; 

$key = q/{{bf{k}+d{{bf{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1469.png"
 ALT="${\bf k} + d{\bf k}$">|; 

$key = q/y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="$y)$">|; 

$key = q/displaystylefrac{3}{8}left(frac{9pi}{8}right)^{1slash3}!hbar,cleft(frac{M}{m_p}right)^{4slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1626.png"
 ALT="$\displaystyle \frac{3}{8}\left(\frac{9\pi}{8}\right)^{1/3} \!\hbar c\left(\frac{M}{m_p}
\right)^{4/3},$">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="$y$">|; 

$key = q/{displaymath}frac{dT}{dz}=-frac{gamma-1}{gamma}frac{mu,g}{R}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img841.png"
 ALT="\begin{displaymath}
\frac{dT}{dz} = -\frac{\gamma -1}{\gamma} \frac{\mu  g}{R}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}S_{{rm{reservoir}=ointfrac{{mathchar'26mskip-12mud}Q}{T_1}=-frac{q}{T_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img872.png"
 ALT="\begin{displaymath}
{\mit\Delta} S_{\rm reservoir} = \oint \frac{{\mathchar'26\mskip-12mud}Q}{T_1} = -\frac{q}{T_1},
\end{displaymath}">|; 

$key = q/{displaymath}{calP}(n)simeqP_N(n)=frac{N!}{n!,(N-n)!},p^n,q^{N-n}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="\begin{displaymath}
{\cal P}(n) \simeq P_N(n) = \frac{N!}{n! (N-n)!} p^n q^{N-n}.
\end{displaymath}">|; 

$key = q/19,000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img830.png"
 ALT="$19,000$">|; 

$key = q/{displaymath}E_n=(n+1slash2),hbar,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1090.png"
 ALT="\begin{displaymath}
E_n = (n + 1/2)  \hbar  \omega,
\end{displaymath}">|; 

$key = q/10^{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img537.png"
 ALT="$10^{12}$">|; 

$key = q/{displaymath}P_r=frac{exp(-beta,E_r)}{sum_rexp(-beta,E_r)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img933.png"
 ALT="\begin{displaymath}
P_r = \frac{\exp(-\beta  E_r)}{\sum_r \exp(-\beta  E_r)}.
\end{displaymath}">|; 

$key = q/lnP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img936.png"
 ALT="$\ln P$">|; 

$key = q/displaystyleC_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1574.png"
 ALT="$\displaystyle C_V$">|; 

$key = q/{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img550.png"
 ALT="${\mit\Omega}$">|; 

$key = q/{displaymath}bar{n}_s=frac{1}{{{rm{e}^{,alpha+beta,epsilon_s}+1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1410.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_s}+ 1}.
\end{displaymath}">|; 

$key = q/{displaymath}p,dV=nuR,dT.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img763.png"
 ALT="\begin{displaymath}
p dV = \nu R dT.
\end{displaymath}">|; 

$key = q/{displaymath}overline{E^2}=frac{1}{Z}frac{partial^2Z}{partialbeta^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img984.png"
 ALT="\begin{displaymath}
\overline{ E^2} = \frac{1}{Z} \frac{\partial^2 Z}{\partial \beta^2}.
\end{displaymath}">|; 

$key = q/V^{(0)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img563.png"
 ALT="$V^{(0)}$">|; 

$key = q/dS=,{mathchar'26mskip-12mud}QslashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img655.png"
 ALT="$dS= {\mathchar'26\mskip-12mud}Q/T$">|; 

$key = q/lnZ_s(N);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1403.png"
 ALT="$\ln Z_s(N)$">|; 

$key = q/{displaymath}lnZ=N,ln!left(sum_r{{rm{e}^{-beta,epsilon_r}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1440.png"
 ALT="\begin{displaymath}
\ln Z = N \ln\!\left(\sum_r {\rm e}^{-\beta \epsilon_r}\right).
\end{displaymath}">|; 

$key = q/(p+q);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$(p+q)$">|; 

$key = q/{displaymath}w=q.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img869.png"
 ALT="\begin{displaymath}
w = q.
\end{displaymath}">|; 

$key = q/{displaymath}frac{dp}{p}=frac{gamma}{gamma-1}frac{dT}{T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img839.png"
 ALT="\begin{displaymath}
\frac{dp}{p} = \frac{\gamma}{\gamma -1} \frac{dT}{T}.
\end{displaymath}">|; 

$key = q/displaystylenuint_0^Tc_V(T,V),dT+E(0,V),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1109.png"
 ALT="$\displaystyle \nu \int_0^T c_V(T, V)  dT + E(0,V),$">|; 

$key = q/deltap;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="$\delta p$">|; 

$key = q/hbar,omega,simeq3,k,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1500.png"
 ALT="$\hbar \omega \simeq 3  k T$">|; 

$key = q/overline{M}_0proptoB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img969.png"
 ALT="$\overline{M}_0 \propto B$">|; 

$key = q/{displaymath}epsilon=p,c,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1480.png"
 ALT="\begin{displaymath}
\epsilon = p c,
\end{displaymath}">|; 

$key = q/{displaymath}p=overline{left(frac{n_1}{N}right)}=frac{overline{n_1}}{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="\begin{displaymath}
p = \overline{\left(\frac{n_1}{N}\right)} = \frac{\overline{n_1}}{N}.
\end{displaymath}">|; 

$key = q/I_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1156.png"
 ALT="$I_z$">|; 

$key = q/n_{i1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="$n_{i1}$">|; 

$key = q/overline{n_1}gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$\overline{n_1}\gg 1$">|; 

$key = q/{displaymath}overline{n_1}=sum_{n_1=0}^NP_N(n_1),n_1=sum_{n_1=0}^Nfrac{N!}{n_1!,(N-n_1)!},p^{n_1},q^{N-n_1},n_1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="434" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="\begin{displaymath}
\overline{n_1} = \sum_{n_1=0}^N P_N(n_1) n_1 = \sum_{n_1=0}^N
\frac{N!}{n_1! (N-n_1)!} p^{n_1} q^{N-n_1}  n_1.
\end{displaymath}">|; 

$key = q/X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$X$">|; 

$key = q/J=7slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$J=7/2$">|; 

$key = q/tilde{E}slashf;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img649.png"
 ALT="$\tilde{E}/f$">|; 

$key = q/A-B>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1630.png"
 ALT="$A-B&gt;0$">|; 

$key = q/displaystyleP(XotimesY);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$\displaystyle P(X \otimes Y)$">|; 

$key = q/{displaymath}W_{if}=-{mitDelta}bar{E}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="\begin{displaymath}
W_{if} = - {\mit\Delta} \bar{E}.
\end{displaymath}">|; 

$key = q/kln(tilde{E}slashsqrt{f},deltaE);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img648.png"
 ALT="$k\ln(\tilde{E}/\sqrt{f}
 \delta E)$">|; 

$key = q/displaystyleepsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1476.png"
 ALT="$\displaystyle \epsilon$">|; 

$key = q/{displaymath}frac{c_V}{T}=gamma+A,T^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1589.png"
 ALT="\begin{displaymath}
\frac{c_V}{T} = \gamma + A T^2.
\end{displaymath}">|; 

$key = q/{displaymath}lambda(tilde{E})simfrac{f}{tilde{E}^2}>0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="\begin{displaymath}
\lambda(\tilde{E}) \sim \frac{f}{\tilde{E}^2}&gt;0.
\end{displaymath}">|; 

$key = q/{mitOmega}equiv{mitOmega}(E,x_1,cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img556.png"
 ALT="${\mit\Omega}\equiv
{\mit\Omega}(E, x_1,\cdots,$">|; 

$key = q/rho({{bf{k});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1471.png"
 ALT="$\rho({\bf k})$">|; 

$key = q/{displaymath}P_N(n_1)=frac{N!}{n_1!,(N-n_1)!},p^{n_1},q^{N-n_1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="\begin{displaymath}
P_N(n_1) = \frac{N!}{n_1 ! (N-n_1)!}  p^{n_1} q^{N-n_1}.
\end{displaymath}">|; 

$key = q/{displaymath}overline{X}=frac{1}{beta}frac{partiallnZ}{partialx}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img995.png"
 ALT="\begin{displaymath}
\overline{X} = \frac{1}{\beta} \frac{\partial \ln Z}{\partial x}.
\end{displaymath}">|; 

$key = q/(fslashbar{E}),bar{X};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img552.png"
 ALT="$(f/\bar{E}) \bar{X}$">|; 

$key = q/dS^{(0)}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img677.png"
 ALT="$dS^{(0)} =0$">|; 

$key = q/{displaymath}{mitPsi}_{s_1,cdots,s_N}(Q_1,Q_2,cdots,Q_N).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1344.png"
 ALT="\begin{displaymath}
{\mit\Psi}_{s_1,\cdots,s_N} (Q_1, Q_2, \cdots, Q_N).
\end{displaymath}">|; 

$key = q/{displaymath}W={{bf{F}!cdot!{{bf{x}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="\begin{displaymath}
W = {\bf F}\!\cdot \!{\bf x}
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{infty}g(v_z),dv_z=n.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1301.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} g(v_z)  dv_z = n.
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{e}^{,alpha+beta,epsilon_r}gg1{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="29" BORDER="0"
 SRC="|."$dir".q|img1449.png"
 ALT="\begin{displaymath}
{\rm e}^{ \alpha+\beta \epsilon_r}\gg 1
\end{displaymath}">|; 

$key = q/{displaymath}ln{calP}(overline{n_1}+eta)simeqln{calP}(overline{n_1})-frac{eta^2}{2,({mitDelta}^astn_1)^2}+cdots.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="\begin{displaymath}
\ln{\cal P}(\overline{n_1}+\eta) \simeq \ln{\cal P}(\overline{n_1}) -
\frac{\eta^2}{2 ({\mit\Delta}^\ast n_1)^2} +\cdots.
\end{displaymath}">|; 

$key = q/kf;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img640.png"
 ALT="$kf$">|; 

$key = q/{mitOmega}equiv{mitOmega}(E,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img538.png"
 ALT="${\mit\Omega} \equiv {\mit\Omega}(E, x)$">|; 

$key = q/P(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="$P(E)$">|; 

$key = q/{mitDelta}^astE=tilde{E}slashsqrt{f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img644.png"
 ALT="${\mit\Delta}^\ast
E = \tilde{E}/\sqrt{f}$">|; 

$key = q/{displaymath}(beta_i-beta_i'),Q>0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img521.png"
 ALT="\begin{displaymath}
(\beta_i - \beta_i') Q &gt;0,
\end{displaymath}">|; 

$key = q/(5slash2),R=20.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1175.png"
 ALT="$(5/2) R= 20.8$">|; 

$key = q/4.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img827.png"
 ALT="$4.4$">|; 

$key = q/X_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img721.png"
 ALT="$X_\alpha$">|; 

$key = q/g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="$g$">|; 

$key = q/beta,epsilon_rllalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1447.png"
 ALT="$\beta \epsilon_r\ll \alpha$">|; 

$key = q/v_{rmmp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1324.png"
 ALT="$v_{\rm mp}$">|; 

$key = q/{displaymath}P_{18}=frac{N_{18}}{N_{{rm{students}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="\begin{displaymath}
P_{18} =
\frac{N_{18}}{N_{\rm students}},
\end{displaymath}">|; 

$key = q/pminfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$\pm \infty$">|; 

$key = q/z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="$z)$">|; 

$key = q/X_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$X_i$">|; 

$key = q/{displaymath}overline{f(u)+g(u)}=sum_{i=1}^{M}P(u_i),[f(u_i)+g(u_i)]=sum_{i=1}^{M}P(u_i),f(u_i)+sum_{i=1}^{M}P(u_i),g(u_i),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="576" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="\begin{displaymath}
\overline{f(u)+g(u)} = \sum_{i=1}^{M}P(u_i) [f(u_i)+g(u_i)]
= \sum_{i=1}^{M}P(u_i) f(u_i)+ \sum_{i=1}^{M}P(u_i) g(u_i),
\end{displaymath}">|; 

$key = q/c_pslashc_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img767.png"
 ALT="$c_p/ c_V$">|; 

$key = q/{displaymath}overline{v_z^{~2}}=frac{k,T}{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1303.png"
 ALT="\begin{displaymath}
\overline{v_z^{&nbsp;2}} = \frac{k T}{m}.
\end{displaymath}">|; 

$key = q/yll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img950.png"
 ALT="$y\ll 1$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=dE+p,dV=nu,c_V,dT+p,dV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img762.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = dE + p dV = \nu  c_V  dT + p  dV.
\end{displaymath}">|; 

$key = q/C^{N}_{n_1,,N-n_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$C^{ N}_{ n_1,  N-n_1}$">|; 

$key = q/{displaymath}f_D(y)rightarrowfrac{3}{y^3}int_0^yx^2,dx=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1259.png"
 ALT="\begin{displaymath}
f_D(y) \rightarrow \frac{3}{y^3} \int_0^y x^2 dx = 1.
\end{displaymath}">|; 

$key = q/displaystyleleft(p,frac{partial}{partialp}right)^2sum_{n_1=0}^Nfrac{N!}{n_1!,(N-n_1)!},p^{n_1}q^{N-n_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$\displaystyle \left(p \frac{\partial}{\partial p}\right)^2\sum_{n_1=0}^N
\frac{N!}{n_1! (N-n_1)!} p^{n_1}q^{N-n_1}$">|; 

$key = q/{displaymath}E=frac{hbar^2}{2,I},J(J+1),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1162.png"
 ALT="\begin{displaymath}
E = \frac{\hbar^2}{2  I}  J(J+1),
\end{displaymath}">|; 

$key = q/beta_iequivbeta(E_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img522.png"
 ALT="$\beta_i\equiv \beta(E_i)$">|; 

$key = q/displaystylefrac{partialln{mitOmega}}{partialE},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="$\displaystyle \frac{\partial \ln {\mit\Omega}}{\partial E},$">|; 

$key = q/d^3{{bf{r},d^3{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1273.png"
 ALT="$d^3{\bf r} d^3{\bf p}$">|; 

$key = q/{figure}{epsfysize{=4.5in{setbox{0=hbox{{epsffile{spin.eps}{}{centerline{rotu0}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="521" BORDER="0"
 SRC="|."$dir".q|img970.png"
 ALT="\begin{figure}
\epsfysize =4.5in\setbox 0=\hbox{\epsffile{spin.eps}}
\centerline{\rotu0}
\end{figure}">|; 

$key = q/{displaymath}tanhysimeqy-frac{y^3}{3}+cdots,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img957.png"
 ALT="\begin{displaymath}
\tanh y \simeq y - \frac{y^3}{3} +\cdots,
\end{displaymath}">|; 

$key = q/{displaymath}C_Vsimeqfrac{3}{2},N,k,frac{k,T}{mu}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1581.png"
 ALT="\begin{displaymath}
C_V \simeq \frac{3}{2} N k \frac{k T}{\mu}.
\end{displaymath}">|; 

$key = q/{displaymath}overline{mu}_Bsimeqmu.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img962.png"
 ALT="\begin{displaymath}
\overline{\mu}_B \simeq \mu.
\end{displaymath}">|; 

$key = q/{displaymath}T=T_0left(1-frac{gamma-1}{gamma}frac{z}{z_0}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img850.png"
 ALT="\begin{displaymath}
T = T_0 \left(1 - \frac{\gamma -1}{\gamma} \frac{z}{z_0} \right),
\end{displaymath}">|; 

$key = q/{{mitSigma}}otimes{{mitSigma}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="${{\mit\Sigma}}\otimes {{\mit\Sigma}}$">|; 

$key = q/x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="$x_1$">|; 

$key = q/{displaymath}q_2=nu,R,T_2,lnfrac{V_c}{V_d}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img898.png"
 ALT="\begin{displaymath}
q_2 = \nu  R  T_2  \ln\frac{V_c}{V_d}.
\end{displaymath}">|; 

$key = q/{displaymath}tilde{n}=N,p=overline{n_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="\begin{displaymath}
\tilde{n} = N p = \overline{n_1},
\end{displaymath}">|; 

$key = q/{mitDelta}^astE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img645.png"
 ALT="${\mit\Delta}^\ast E$">|; 

$key = q/{displaymath}F(v),dv=intf({{bf{v}),d^3{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1313.png"
 ALT="\begin{displaymath}
F(v) dv = \int f({\bf v})  d^3{\bf v},
\end{displaymath}">|; 

$key = q/displaystyleleft(p,frac{partial}{partialp}right)^2(p+q)^N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$\displaystyle \left(p \frac{\partial}{\partial p}\right)^2(p+q)^N$">|; 

$key = q/{displaymath}frac{partial{mitOmega}}{partialx}=frac{partial({mitOmega}bar{X})}{partialE},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img549.png"
 ALT="\begin{displaymath}
\frac{\partial {\mit\Omega}}{\partial x} = \frac{\partial({\mit\Omega} \bar{X})}{\partial E},
\end{displaymath}">|; 

$key = q/F=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1539.png"
 ALT="$F=1/2$">|; 

$key = q/{displaymath}(c_V)_{{rm{vibration}=(n-1),R.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1166.png"
 ALT="\begin{displaymath}
(c_V)_{\rm vibration} = (n-1)  R.
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}_f-{mitOmega}_iequiv{mitDelta}{mitOmega}geq0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img617.png"
 ALT="\begin{displaymath}
{\mit\Omega}_f -{\mit\Omega}_i \equiv {\mit\Delta} {\mit\Omega}\geq 0,
\end{displaymath}">|; 

$key = q/P(X);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$P(X)$">|; 

$key = q/y_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$y_i)$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}GequivX'(x,y),dx+Y'(x,y),dz,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}G \equiv X'(x,y)  dx + Y'(x,y)  dz,
\end{displaymath}">|; 

$key = q/{displaymath}lnZ=Nlnzeta-lnN!,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1056.png"
 ALT="\begin{displaymath}
\ln Z = N \ln \zeta - \ln N!,
\end{displaymath}">|; 

$key = q/{mitOmega}^{(0)}equiv{mitOmega}^{(0)}(E,x_1,cdots,x_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img566.png"
 ALT="${\mit\Omega}^{(0)}\equiv {\mit\Omega}^{(0)}(E, x_1, \cdots, x_n)$">|; 

$key = q/E^{(0)}-E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="$E^{(0)}- E$">|; 

$key = q/f({{bf{r},{{bf{v}),d^3{{bf{r},d^3{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1278.png"
 ALT="$f({\bf r}, {\bf v}) d^3{\bf r} d^3{\bf v}$">|; 

$key = q/P_r(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="$P_r(t)$">|; 

$key = q/{displaymath}T_1,V_a^{,gamma-1}=T_2,V_d^{,gamma-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img901.png"
 ALT="\begin{displaymath}
T_1  V_a^{ \gamma -1}= T_2  V_d^{ \gamma -1}.
\end{displaymath}">|; 

$key = q/V^N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img719.png"
 ALT="$V^N$">|; 

$key = q/{displaymath}frac{dH}{dt}=-frac{1}{2}sum_rsum_sW_{rs},(P_r-P_s),(lnP_r-lnP_s).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="\begin{displaymath}
\frac{d H}{dt}= -\frac{1}{2}\sum_r \sum_s W_{rs} (P_r-P_s) (\ln P_r
-\ln P_s).
\end{displaymath}">|; 

$key = q/A^{(0)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="$A^{(0)}$">|; 

$key = q/V+dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="$V + dV$">|; 

$key = q/displaystylefrac{3}{20}left(frac{9pi}{8}right)^{2slash3}frac{hbar^2}{m}left(frac{M}{m_p}right)^{5slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1610.png"
 ALT="$\displaystyle \frac{3}{20}\left(\frac{9\pi}{8}\right)^{2/3}\frac{\hbar^2}{m}
\left(\frac{M}{m_p}\right)^{5/3},$">|; 

$key = q/p+q=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$p+q=1$">|; 

$key = q/10^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img865.png"
 ALT="$10^\circ$">|; 

$key = q/Ngg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$N\gg 1$">|; 

$key = q/p^{,1-gamma},T^{,gamma}=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img852.png"
 ALT="$p^{ 1-\gamma}  T^{ \gamma} =$">|; 

$key = q/P_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1520.png"
 ALT="$P_\oplus$">|; 

$key = q/{displaymath}P(XmidY)=~_{lt,{mitOmega}({mitSigma})rightarrowinfty}frac{{mitOmega}(XmidY)}{{mitOmega}({mitSigma})},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="\begin{displaymath}
P(X\mid Y) =&nbsp;_{lt {\mit\Omega}({\mit\Sigma})\rightarrow\infty}
\frac{
{\mit\Omega}(X \mid Y)}{{\mit\Omega}({\mit\Sigma})},
\end{displaymath}">|; 

$key = q/{displaymath}E=frac{p^2}{2,m}+frac{1}{2}kappa,x^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1083.png"
 ALT="\begin{displaymath}
E = \frac{p^2}{2 m} + \frac{1}{2}\kappa  x^2,
\end{displaymath}">|; 

$key = q/partialoverline{E}slashpartialTgeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img988.png"
 ALT="$\partial \overline{E}/\partial T \geq 0$">|; 

$key = q/p(z+dz),A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img804.png"
 ALT="$p(z+dz) A$">|; 

$key = q/z_1simeq3.5,z_0simeq29.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img864.png"
 ALT="$z_1 \simeq 3.5 z_0 \simeq 29.4$">|; 

$key = q/{displaymath}{mitOmega}^ast(E)=frac{delta^astE}{deltaE},{mitOmega}(E).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img633.png"
 ALT="\begin{displaymath}
{\mit\Omega}^\ast(E) = \frac{\delta^\ast E}{\delta E} {\mit\Omega}(E).
\end{displaymath}">|; 

$key = q/displaystylefrac{int_{-infty}^{infty}exp[-beta,(epsilon_i+E')],epsilon_i,dq_1cdo{int_{-infty}^{infty}exp[-beta,(epsilon_i+E')],dq_1cdotsdp_f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1073.png"
 ALT="$\displaystyle \frac{
\int_{-\infty}^{\infty} \exp[-\beta (\epsilon_i + E')] 
...
...p_f}
{\int_{-\infty}^{\infty} \exp[-\beta (\epsilon_i + E')] dq_1\cdots dp_f}$">|; 

$key = q/overline{n_1}pm3,{mitDelta}^astn_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$\overline{n_1}\pm 3 {\mit\Delta}^\ast n_1$">|; 

$key = q/m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$m$">|; 

$key = q/{displaymath}dlnZ=beta,,{mathchar'26mskip-12mud}W-d(overline{E},beta)+beta,doverline{E},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1003.png"
 ALT="\begin{displaymath}
d \ln Z = \beta   {\mathchar'26\mskip-12mud}W - d(\overline{E}  \beta) +\beta  d\overline{E},
\end{displaymath}">|; 

$key = q/S(bar{E},bar{x}_alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img584.png"
 ALT="$S(\bar{E}, \bar{x}_\alpha)$">|; 

$key = q/displaystyleQ';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img514.png"
 ALT="$\displaystyle Q'$">|; 

$key = q/h=6.61times10^{-34},{{rm{J,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1461.png"
 ALT="$h=6.61\times 10^{-34} {\rm J s}$">|; 

$key = q/{displaymath}E'-tilde{E}'=-(E-tilde{E})=-eta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="\begin{displaymath}
E' - \tilde{E}' = - (E-\tilde{E}) = -\eta.
\end{displaymath}">|; 

$key = q/bar{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="$\bar{p}$">|; 

$key = q/X';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="$X'$">|; 

$key = q/c_V^{(L)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1586.png"
 ALT="$c_V^{(L)}$">|; 

$key = q/{displaymath}M_C=frac{15}{64},(5pi)^{1slash2},frac{(hbar,cslashG)^{1slash2}}{m_p^{,2}}=1.72,M_odot{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1634.png"
 ALT="\begin{displaymath}
M_C = \frac{15}{64} (5\pi)^{1/2}  \frac{(\hbar c/G)^{1/2}}{m_p^{ 2}}=
1.72 M_\odot
\end{displaymath}">|; 

$key = q/{displaymath}sum_{n_s=0}^infty{{rm{e}^{-beta,n_s,epsilon_s}=1+{{rm{e}^{-beta,epsn_s}+cdots=frac{1}{1-{{rm{e}^{-beta,epsilon_s}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1422.png"
 ALT="\begin{displaymath}
\sum_{n_s=0}^\infty {\rm e}^{-\beta n_s \epsilon_s}
= 1 +...
...psilon_s} +
\cdots = \frac{1}{1-{\rm e}^{-\beta \epsilon_s}}.
\end{displaymath}">|; 

$key = q/J=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img971.png"
 ALT="$J=1/2$">|; 

$key = q/{displaymath}zeta=frac{V}{h_0^{~3}}intexp[-(betaslash2m),p^2],d^3{{bf{p}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1025.png"
 ALT="\begin{displaymath}
\zeta = \frac{V}{h_0^{&nbsp;3}}\int \exp[-(\beta/2m) p^2] d^3{\bf p}
\end{displaymath}">|; 

$key = q/R=8.3143,{{rm{joulesslashmoleslashdeg.};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img783.png"
 ALT="$R= 8.3143 {\rm joules/mole/deg.}$">|; 

$key = q/{{bf{r}+d{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1271.png"
 ALT="${\bf r} + d{\bf r}$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="$T$">|; 

$key = q/{displaymath}f_D(y)equivfrac{3}{y^3}int_0^yfrac{expx}{(expx-1)^2},x^4,dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1253.png"
 ALT="\begin{displaymath}
f_D(y) \equiv \frac{3}{y^3}\int_0^y \frac{\exp x}
{(\exp x -1)^2} x^4 dx.
\end{displaymath}">|; 

$key = q/pproptoT^{,gammaslash(gamma-1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img853.png"
 ALT="$p \propto T^{ \gamma/(\gamma -1)}$">|; 

$key = q/bar{p},,d{{bf{S}_i!cdot!d{{bf{x}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="$\bar{p}  d{\bf S}_i
\!\cdot\! d{\bf x}_i$">|; 

$key = q/ln{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img577.png"
 ALT="$\ln{\mit\Omega}$">|; 

$key = q/{{bf{p}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="${\bf p}_i$">|; 

$key = q/displaystyle{{rm{for~}omega>omega_D.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1239.png"
 ALT="$\displaystyle {\rm for&nbsp;} \omega&gt;\omega_D.$">|; 

$key = q/{displaymath}c_s=sqrt{frac{gamma,k,T}{m}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1336.png"
 ALT="\begin{displaymath}
c_s=\sqrt{\frac{\gamma  k T}{m}},
\end{displaymath}">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=T,dS,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img715.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = T dS,
\end{displaymath}">|; 

$key = q/{displaymath}F(v),dv=4pif(v),v^2,dv,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1320.png"
 ALT="\begin{displaymath}
F(v)  dv = 4\pi f(v)  v^2 dv,
\end{displaymath}">|; 

$key = q/{displaymath}dS=frac{{mathchar'26mskip-12mud}Q}{T},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img653.png"
 ALT="\begin{displaymath}
dS =\frac{{\mathchar'26\mskip-12mud}Q}{T},
\end{displaymath}">|; 

$key = q/displaystylepleft[N+p,N,(N-1)right]=N,pleft[1+p,N-pright];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$\displaystyle p\left[N+p N (N-1)\right]= N p\left[1+p N-p\right]$">|; 

$key = q/{displaymath}dS=frac{dE+bar{p},dV}{T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img668.png"
 ALT="\begin{displaymath}
d S = \frac{ dE + \bar{p} dV}{T}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}E=Q-W,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="\begin{displaymath}
{\mit\Delta} E = Q - W,
\end{displaymath}">|; 

$key = q/^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$^\circ $">|; 

$key = q/C';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img923.png"
 ALT="$C'$">|; 

$key = q/displaystylenleft(frac{m}{2pi,k,T}right)^{3slash2}int_{(v_x)}int_{(v_y)}exp[-(mslash2,k,T)(v_x^{~2}+v_y^{~2}+v_z^{~2})],dv_x,dv_y,dv_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="551" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1297.png"
 ALT="$\displaystyle n \left(\frac{m}{2\pi  k T}\right)^{3/2}
\int_{(v_x)} \int_{(v_y)} \exp[- (m/2  k T)(v_x^{&nbsp;2}+v_y^{&nbsp;2}+v_z^{&nbsp;2})] 
dv_x dv_y  dv_z$">|; 

$key = q/{displaymath}L_odot=4pi,R_odot^{~2},sigma,T_odot^{~4},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1512.png"
 ALT="\begin{displaymath}
L_\odot = 4\pi  R_\odot^{&nbsp;2}   \sigma  T_\odot^{&nbsp;4},
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialln{mitOmega}}{partialE}=beta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img938.png"
 ALT="\begin{displaymath}
\frac{\partial \ln {\mit\Omega}}{\partial E} = \beta.
\end{displaymath}">|; 

$key = q/dHslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$d H/dt$">|; 

$key = q/32,000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img835.png"
 ALT="$32,000$">|; 

$key = q/{displaymath}bar{E}_f+bar{E}_f'=E_i+E_i'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img511.png"
 ALT="\begin{displaymath}
\bar{E}_f +\bar{E}_f' = E_i + E_i'.
\end{displaymath}">|; 

$key = q/{displaymath}S=2,klnfrac{{mitOmega}_f}{{mitOmega}_i}=2,N,klnfrac{V_f}{V_i}=2,nu,R,ln2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1053.png"
 ALT="\begin{displaymath}
S = 2  k\ln \frac{{\mit\Omega}_f}{{\mit\Omega}_i} = 2 N  k\ln \frac{V_f}{V_i} =
2 \nu  R  \ln 2.
\end{displaymath}">|; 

$key = q/eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img885.png"
 ALT="$\eta$">|; 

$key = q/{displaymath}k_F=left(3,pi^2,frac{N}{V}right)^{1slash3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1560.png"
 ALT="\begin{displaymath}
k_F = \left(3 \pi^2 \frac{N}{V}\right)^{1/3}.
\end{displaymath}">|; 

$key = q/{displaymath}|{mitPsi}_{s_1,cdots,s_N}(Q_1,Q_2,cdots,Q_N)|^2,dQ_1,dQ_2,cdots,dQ_N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1346.png"
 ALT="\begin{displaymath}
\vert{\mit\Psi}_{s_1,\cdots,s_N} (Q_1, Q_2, \cdots, Q_N)\vert^2 dQ_1 dQ_2 \cdots dQ_N.
\end{displaymath}">|; 

$key = q/{displaymath}p=p_0expleft(-frac{z}{z_0}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img857.png"
 ALT="\begin{displaymath}
p = p_0\exp\left(-\frac{z}{z_0}\right),
\end{displaymath}">|; 

$key = q/{displaymath}bar{n}_s=frac{sum_{n_s}n_s,{{rm{e}^{-beta,n_s,epsilon_s},sum_{n_1,ne}^{-beta,(n_1,epsilon_1+n_2,epsilon_2+cdots)}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="374" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img1389.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{\sum_{n_s} n_s {\rm e}^{-\beta n_s \eps...
... {\rm e}^{-\beta (n_1 \epsilon_1+n_2 \epsilon_2+
\cdots)}}.
\end{displaymath}">|; 

$key = q/{displaymath}dS^{(0)}=dS+dS'=left(frac{1}{T}-frac{1}{T'}right),dE+left(frac{bar{p}}{T}-frac{bar{p}'}{T'}right),dV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img676.png"
 ALT="\begin{displaymath}
dS^{(0)} = dS + dS' = \left(\frac{1}{T}-\frac{1}{T'}\right) dE+
\left(\frac{\bar{p}}{T}-\frac{\bar{p}'}{T'}\right) dV.
\end{displaymath}">|; 

$key = q/{displaymath}c_V=3,Rleft(frac{theta_E}{T}right)^2frac{exp(theta_EslashT)}{[exp(theta_EslashT)-1]^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1190.png"
 ALT="\begin{displaymath}
c_V = 3 R \left(\frac{\theta_E}{T}\right)^2 \frac{\exp(\theta_E / T)}
{[\exp(\theta_E/T) - 1]^2}.
\end{displaymath}">|; 

$key = q/{{bf{q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="${\bf q}$">|; 

$key = q/6times5times4times3times2times1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$6\times 5\times 4 \times 3 \times 2 \times 1$">|; 

$key = q/{displaymath}frac{A}{B}>1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1632.png"
 ALT="\begin{displaymath}
\frac{A}{B} &gt; 1.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{E}={{bf{E}_0exp[,{{rm{i},({{bf{k}!cdot!{{bf{r}-omega,t)],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1463.png"
 ALT="\begin{displaymath}
{\bf E} = {\bf E}_0 \exp[ {\rm i} ({\bf k}\!\cdot \!{\bf r} - \omega  t)],
\end{displaymath}">|; 

$key = q/{displaymath}E=frac{hbar^2pi^2}{2,m,L^2}left(n_1^{~2}+n_2^{~2}+n_3^{~2}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1146.png"
 ALT="\begin{displaymath}
E = \frac{\hbar^2 \pi^2}{2  m  L^2} \left(n_1^{&nbsp;2}+n_2^{&nbsp;2}+ n_3^{&nbsp;2}\right),
\end{displaymath}">|; 

$key = q/v_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1286.png"
 ALT="$v_z$">|; 

$key = q/S=kln{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img635.png"
 ALT="$S=k\ln {\mit\Omega}$">|; 

$key = q/{displaymath}{mitOmega}(E)sim1+C,(E-E_0)^f,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img697.png"
 ALT="\begin{displaymath}
{\mit\Omega}(E) \sim 1 + C  (E - E_0)^f,
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{2},N,k.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1575.png"
 ALT="$\displaystyle \frac{3}{2} N k.$">|; 

$key = q/{displaymath}lnZ=-N,lnN+N+N,ln!left({sum_r{{rm{e}^{-beta,epsilon_r}}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1455.png"
 ALT="\begin{displaymath}
\ln Z = -N \ln N + N +N \ln\! \left({\sum_r {\rm e}^{-\beta \epsilon_r}}\right).
\end{displaymath}">|; 

$key = q/{displaymath}epsilon=frac{{{bf{p}^2}{2,m}+epsilon^{{rm{int},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1267.png"
 ALT="\begin{displaymath}
\epsilon = \frac{{\bf p}^2}{2 m} + \epsilon^{\rm int},
\end{displaymath}">|; 

$key = q/C=C(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$C=C(\theta)$">|; 

$key = q/1slash6+1slash6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$1/6+1/6$">|; 

$key = q/Tggtheta_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1192.png"
 ALT="$T\gg \theta_E$">|; 

$key = q/{displaymath}T,V^{,gamma-1}={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img899.png"
 ALT="\begin{displaymath}
T  V^{ \gamma -1} = {\rm constant}.
\end{displaymath}">|; 

$key = q/k,Tslashmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1579.png"
 ALT="$k T/\mu$">|; 

$key = q/{bf{r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1270.png"
 ALT="$\bf r$">|; 

$key = q/{displaymath}L_oplus=4pi,R_oplus^{~2},sigma,T_oplus^{~4},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1517.png"
 ALT="\begin{displaymath}
L_\oplus = 4\pi  R_\oplus^{&nbsp;2}   \sigma  T_\oplus^{&nbsp;4},
\end{displaymath}">|; 

$key = q/{displaymath}c_V=12.47,,{{rm{joulesslashmoleslashdeg.}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img785.png"
 ALT="\begin{displaymath}
c_V = 12.47   {\rm joules/mole/deg.}
\end{displaymath}">|; 

$key = q/{displaymath}ln{mitOmega}'(E')=ln{mitOmega}'(tilde{E}')+beta'(tilde{E}'),(-eta)-frac{1}{2},lambda'(tilde{E}'),(-eta)^2+cdots,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="477" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="\begin{displaymath}
\ln{\mit\Omega}'(E') = \ln{\mit\Omega}'(\tilde{E}')+\beta'(\...
...\eta)
-\frac{1}{2} 
\lambda'(\tilde{E}') (-\eta)^2
+\cdots,
\end{displaymath}">|; 

$key = q/d{{bf{x}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="$d{\bf x}_i$">|; 

$key = q/{displaymath}K=frac{1}{2,m}(p_x^{~2}+p_y^{~2}+p_z^{~2}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1144.png"
 ALT="\begin{displaymath}
K = \frac{1}{2 m} (p_x^{&nbsp;2}+ p_y^{&nbsp;2}+p_z^{&nbsp;2}).
\end{displaymath}">|; 

$key = q/c_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img772.png"
 ALT="$c_V$">|; 

$key = q/A^{(0)}=A+A';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="$A^{(0)} = A + A'$">|; 

$key = q/3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1231.png"
 ALT="$3$">|; 

$key = q/{figure}{epsfysize{=3in{setbox{0=hbox{{epsffile{eff.eps}{}{centerline{rotu0}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="542" HEIGHT="349" BORDER="0"
 SRC="|."$dir".q|img1341.png"
 ALT="\begin{figure}
\epsfysize =3in\setbox 0=\hbox{\epsffile{eff.eps}}
\centerline{\rotu0}
\end{figure}">|; 

$key = q/(1slash2),k,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img534.png"
 ALT="$(1/2) k T$">|; 

$key = q/{displaymath}eta=frac{673-288}{673}simeq0.57,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img911.png"
 ALT="\begin{displaymath}
\eta = \frac{673-288}{673} \simeq 0.57,
\end{displaymath}">|; 

$key = q/N^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$N^{-1/2}$">|; 

$key = q/{displaymath}lnP_r=lnC'+ln{mitOmega}'(E^{(0)})-left[frac{partialln{mitOmega}'}{partialE'}right]_0E_r+cdots.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img927.png"
 ALT="\begin{displaymath}
\ln P_r = \ln C' +\ln {\mit\Omega}'(E^{(0)}) -\left[\frac{\partial \ln {\mit\Omega}'}
{\partial E'} \right]_0 E_r +\cdots.
\end{displaymath}">|; 

$key = q/ln!10,z_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img824.png"
 ALT="$\ln \!10  z_0$">|; 

$key = q/6,p^2,q^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$6 p^2 q^2$">|; 

$key = q/rhoproptopslashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img823.png"
 ALT="$\rho\propto
p/T$">|; 

$key = q/MllM_odot;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1620.png"
 ALT="$M\ll M_\odot$">|; 

$key = q/L_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1519.png"
 ALT="$L_\oplus$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W=overline{X},dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img993.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W = \overline{X} dx,
\end{displaymath}">|; 

$key = q/E=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$E=$">|; 

$key = q/v_{{rm{mp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$v_{\rm mp}$">|; 

$key = q/{displaymath}P_i=frac{{mitOmega}_i}{{mitOmega}_f}=left(frac{V_i}{V_f}right)^N=left(frac{1}{2}right)^N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img604.png"
 ALT="\begin{displaymath}
P_i = \frac{{\mit\Omega}_i}{{\mit\Omega}_f} = \left(\frac{V_i}{V_f}\right)^N =\left(\frac{1}{2}
\right)^N.
\end{displaymath}">|; 

$key = q/chiproptoT^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img967.png"
 ALT="$\chi\propto T^{-1}$">|; 

$key = q/I_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1155.png"
 ALT="$I_y$">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img810.png"
 ALT="$\mu$">|; 

$key = q/T_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img876.png"
 ALT="$T_2$">|; 

$key = q/m_prightarrowm_pslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1636.png"
 ALT="$m_p\rightarrow m_p/2$">|; 

$key = q/{displaymath}P(XotimesY)=~_{lt,{mitOmega}({mitSigma})rightarrowinfty}frac{{mitOmega}({{mitSigma}}otimes{{mitSigma}})}=P(X),P(Y).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="\begin{displaymath}
P(X\otimes Y) =&nbsp;_{lt {\mit\Omega}({\mit\Sigma})\rightarrow...
...it\Omega}({{\mit\Sigma}}\otimes {{\mit\Sigma}})}= P(X)  P(Y).
\end{displaymath}">|; 

$key = q/I_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1154.png"
 ALT="$I_x$">|; 

$key = q/{displaymath}lnZ=alpha,N-sum_rln,(1-{{rm{e}^{-alpha-beta,epsilon_r}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1434.png"
 ALT="\begin{displaymath}
\ln Z = \alpha N - \sum_r\ln  (1-{\rm e}^{-\alpha-\beta \epsilon_r}),
\end{displaymath}">|; 

$key = q/p=q=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$p=q=1/2$">|; 

$key = q/dyslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="$dy/dx$">|; 

$key = q/C^{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1604.png"
 ALT="$C^{12}$">|; 

$key = q/x_alpharightarrowx_alpha+dx_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$x_\alpha \rightarrow x_\alpha + dx_\alpha$">|; 

$key = q/H_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$H_2$">|; 

$key = q/m,d^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1158.png"
 ALT="$m  d^2$">|; 

$key = q/29,028;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img832.png"
 ALT="$29,028$">|; 

$key = q/{displaymath}dE=left(frac{partialE}{partialT}right)_VdT+left(frac{partialE}{partialV}right)_TdV,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img736.png"
 ALT="\begin{displaymath}
dE = \left(\frac{\partial E}{\partial T}\right)_V dT + \left( \frac{\partial E}
{\partial V}\right)_T dV,
\end{displaymath}">|; 

$key = q/P(u_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$P(u_1)$">|; 

$key = q/{displaymath}P_3(2)=P(1otimes1otimes2mid2otimes1otimes1mid1otimes2otimes1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="\begin{displaymath}
P_3(2)=
P(1\otimes 1\otimes 2 \mid 2\otimes 1\otimes 1 \mid 1\otimes 2 \otimes 1).
\end{displaymath}">|; 

$key = q/5times10^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1137.png"
 ALT="$5\times 10^4$">|; 

$key = q/displaystyleP(X)+P(Y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$\displaystyle P(X) + P(Y),$">|; 

$key = q/{displaymath}K=frac{1}{2},I_x,omega_x^{~2}+frac{1}{2},I_y,omega_y^{~2}+frac{1}{2},I_z,omega_z^{~2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1150.png"
 ALT="\begin{displaymath}
K = \frac{1}{2}  I_x  \omega_x^{&nbsp;2}+\frac{1}{2}  I_y  \omega_y^{&nbsp;2}+
\frac{1}{2}  I_z  \omega_z^{&nbsp;2},
\end{displaymath}">|; 

$key = q/{displaymath}dS^{(0)}=dS+dS'=left(frac{1}{T}-frac{1}{T'}right){mathchar'26mskip-12mud}Q.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img658.png"
 ALT="\begin{displaymath}
dS^{(0)} = dS + dS' = \left( \frac{1}{T}- \frac{1}{T'}\right){\mathchar'26\mskip-12mud}Q.
\end{displaymath}">|; 

$key = q/{displaymath}{mitPsi}=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1368.png"
 ALT="\begin{displaymath}
{\mit\Psi} = 0
\end{displaymath}">|; 

$key = q/{displaymath}bar{n}_s=frac{0+{{rm{e}^{-beta,epsilon_s},Z_s(N-1)+2,{{rm{e}^{-2,be-1)+{{rm{e}^{-2,beta,epsilon_s},Z_s(N-2)+cdots},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1427.png"
 ALT="\begin{displaymath}
\bar{n}_s=\frac{0 +{\rm e}^{-\beta \epsilon_s} Z_s(N-1)
+ ...
...,Z_s(N-1)
+ {\rm e}^{-2 \beta \epsilon_s} Z_s(N-2)+\cdots},
\end{displaymath}">|; 

$key = q/r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$r$">|; 

$key = q/s_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1343.png"
 ALT="$s_i$">|; 

$key = q/p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img815.png"
 ALT="$p_0$">|; 

$key = q/{mitOmega}equiv{mitOmega}(y_1,cdots,y_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img592.png"
 ALT="${\mit\Omega}\equiv {\mit\Omega}(y_1,\cdots, y_n)$">|; 

$key = q/{mitOmega}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img595.png"
 ALT="${\mit\Omega}_i$">|; 

$key = q/{displaymath}gammaequivfrac{c_p}{c_V}=frac{5}{3}=1.667.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img787.png"
 ALT="\begin{displaymath}
\gamma \equiv \frac{c_p}{c_V} = \frac{5}{3} = 1.667.
\end{displaymath}">|; 

$key = q/|{mitPsi}|^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1357.png"
 ALT="$\vert{\mit\Psi}\vert^2$">|; 

$key = q/{displaymath}S=nu,Rleft[lnV-frac{3}{2}lnbeta+frac{3}{2}ln!left(frac{2,pi,m,k}{h_0^{~2}}right)+frac{3}{2}right]+k,(-NlnN+N).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="534" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1059.png"
 ALT="\begin{displaymath}
S = \nu  R\left[ \ln V - \frac{3}{2} \ln \beta +\frac{3}{2}...
...m k}{h_0^{&nbsp;2}}\right) + \frac{3}{2}\right]+ k (-N\ln N + N).
\end{displaymath}">|; 

$key = q/c_V^{(L)}=A,T^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1587.png"
 ALT="$c_V^{(L)}=A T^3$">|; 

$key = q/{displaymath}overline{p}=frac{1}{beta}frac{partiallnZ}{partialV}=frac{1}{beta}frac{N}{V},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1032.png"
 ALT="\begin{displaymath}
\overline{p} = \frac{1}{\beta}\frac{\partial\ln Z}{\partial V} = \frac{1}{\beta}
\frac{N}{V},
\end{displaymath}">|; 

$key = q/displaystylefrac{overline{p^2}}{2,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1085.png"
 ALT="$\displaystyle \frac{\overline{p^2}}{2 m}$">|; 

$key = q/{displaymath}alpha=-lnN+lnleft({sum_r{{rm{e}^{-beta,epsilon_r}}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1454.png"
 ALT="\begin{displaymath}
\alpha =-\ln N + \ln \left({\sum_r {\rm e}^{-\beta \epsilon_r}}\right).
\end{displaymath}">|; 

$key = q/T_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img871.png"
 ALT="$T_1$">|; 

$key = q/{displaymath}int_{({{bf{r})}int_{({{bf{v})}f({{bf{r},{{bf{v}),d^3{{bf{r},d^3{{bf{v}=N:{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1283.png"
 ALT="\begin{displaymath}
\int_{({\bf r})} \int_{({\bf v})} f({\bf r}, {\bf v}) d^3{\bf r} d^3{\bf v} = N:
\end{displaymath}">|; 

$key = q/{displaymath}bar{n}_rll1{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1450.png"
 ALT="\begin{displaymath}
\bar{n}_r \ll 1
\end{displaymath}">|; 

$key = q/pggm,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1621.png"
 ALT="$p\gg m c$">|; 

$key = q/(p_{i,x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="$(p_{i x}$">|; 

$key = q/{displaymath}{mitPsi}(cdotsQ_icdotsQ_jcdots)=-{mitPsi}(cdotsQ_jcdotsQ_icdots).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1363.png"
 ALT="\begin{displaymath}
{\mit\Psi}(\cdots Q_i\cdots Q_j\cdots)=
-{\mit\Psi}(\cdots Q_j\cdots Q_i\cdots).
\end{displaymath}">|; 

$key = q/E_i';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img505.png"
 ALT="$E_i'$">|; 

$key = q/mathaccent"7016relax{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$\mathaccent ''7016\relax {c}$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img894.png"
 ALT="$a$">|; 

$key = q/n=NslashV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1291.png"
 ALT="$n=N/V$">|; 

$key = q/{displaymath}Y',frac{partialsigma}{partialx}=X',frac{partialsigma}{partialy}=frac{X',Y'}{tau},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="\begin{displaymath}
Y'  \frac{\partial \sigma}{\partial x} = X'  \frac{\partial\sigma}{\partial y}
=\frac{X'  Y'}{\tau},
\end{displaymath}">|; 

$key = q/{displaymath}frac{dH}{dt}leq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="\begin{displaymath}
\frac{d H}{dt} \leq 0.
\end{displaymath}">|; 

$key = q/{mitOmega}({mitSigma});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="${\mit\Omega}({\mit\Sigma})$">|; 

$key = q/(-);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img942.png"
 ALT="$(-)$">|; 

$key = q/{displaymath}overline{mu}_B=frac{P_{+},mu+P_{-},(-mu)}{P_{+}+P_{-}}=mu,,frac{expxp(-beta,mu,B)}{exp(beta,mu,B)+exp(-beta,mu,B)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img954.png"
 ALT="\begin{displaymath}
\overline{\mu}_B = \frac{ P_{+}  \mu + P_{-}  (-\mu)}{P_{+...
...  \mu  B)}
{ \exp(\beta  \mu  B)+ \exp(-\beta  \mu  B)}.
\end{displaymath}">|; 

$key = q/{displaymath}lambda_0=lambda(tilde{E})+lambda'(tilde{E}').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="\begin{displaymath}
\lambda_0 = \lambda (\tilde{E})+\lambda'(\tilde{E}').
\end{displaymath}">|; 

$key = q/deltaq_i,deltap_i=hbar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="$\delta q_i  \delta p_i=\hbar$">|; 

$key = q/{displaymath}int_0^inftysigma_D(omega),domega=int_0^{omega_D}sigma_c(omega),domega=3,N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img1240.png"
 ALT="\begin{displaymath}
\int_0^\infty \sigma_D(\omega)  d\omega = \int_0^{\omega_D }\sigma_c(\omega)  d\omega =
3  N.
\end{displaymath}">|; 

$key = q/dV=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img757.png"
 ALT="$dV=0$">|; 

$key = q/displaystylebar{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img680.png"
 ALT="$\displaystyle \bar{p}$">|; 

$key = q/{displaymath}overline{epsilon_i}=frac{int_{-infty}^{infty}exp[-betaE(q_1,cdots,pinfty}exp[-betaE(q_1,cdots,p_f)],dq_1cdotsdp_f}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1071.png"
 ALT="\begin{displaymath}
\overline{\epsilon_i} = \frac{
\int_{-\infty}^{\infty} \exp[...
...}^{\infty} \exp[-\beta E(q_1,\cdots, p_f)] 
dq_1\cdots dp_f}.
\end{displaymath}">|; 

$key = q/{displaymath}c_p=frac{1}{nu}left(frac{{mathchar'26mskip-12mud}Q}{dT}right)_p.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img756.png"
 ALT="\begin{displaymath}
c_p = \frac{1}{\nu}\left(\frac{{\mathchar'26\mskip-12mud}Q}{dT}\right)_p.
\end{displaymath}">|; 

$key = q/displaystyleint_{-infty}^{infty}exp[-(betaslash2m),p_x^{~2}],dp_x,int_{-infty}^{infty}exp[-(betaslash2m),p_y^{~2}],dp_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1027.png"
 ALT="$\displaystyle \int_{-\infty}^{\infty}\exp[-(\beta/2m) p_x^{&nbsp;2}] dp_x 
\int_{-\infty}^{\infty}\exp[-(\beta/2m) p_y^{&nbsp;2}] dp_y $">|; 

$key = q/dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img539.png"
 ALT="$dx$">|; 

$key = q/displaystylenuint_0^Tfrac{c_V(T,V)}{T},dT.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1111.png"
 ALT="$\displaystyle \nu \int_0^T \frac{c_V(T,V)}{T}  dT.$">|; 

$key = q/{displaymath}overline{mu}_Bsimeqfrac{mu^2B}{k,T},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img960.png"
 ALT="\begin{displaymath}
\overline{\mu}_B \simeq \frac{\mu^2 B}{k T},
\end{displaymath}">|; 

$key = q/displaystylefrac{2pi}{L_x},n_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1210.png"
 ALT="$\displaystyle \frac{2\pi}{L_x}  n_x,$">|; 

$key = q/p_{i,z});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="$p_{i z})$">|; 

$key = q/V=V_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img600.png"
 ALT="$V=V_i$">|; 

$key = q/{displaymath}lnZ=Nlnzeta=Nleft[lnV-frac{3}{2}lnbeta+frac{3}{2}ln!left(frac{2,pi,m}{h_0^{~2}}right)right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1031.png"
 ALT="\begin{displaymath}
\ln Z = N\ln \zeta = N \left[\ln V - \frac{3}{2}\ln \beta +\frac{3}{2}
\ln\!\left(\frac{2 \pi  m}{h_0^{&nbsp;2}}\right)\right].
\end{displaymath}">|; 

$key = q/R^{,3N-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="$R^{ 3N-1}$">|; 

$key = q/mu_B=-mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img945.png"
 ALT="$\mu_B = -\mu$">|; 

$key = q/{displaymath}E=epsilon_i(p_i)+E'(q_1,cdots,p_f),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1067.png"
 ALT="\begin{displaymath}
E = \epsilon_i(p_i) + E'(q_1,\cdots, p_f),
\end{displaymath}">|; 

$key = q/{{bf{r}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="${\bf r}_i$">|; 

$key = q/dv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1315.png"
 ALT="$dv$">|; 

$key = q/{displaymath}bar{n}_s=frac{1}{{{rm{e}^{,alpha+beta,epsilon_s}-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1431.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_s}-1}.
\end{displaymath}">|; 

$key = q/{{bf{E}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1464.png"
 ALT="${\bf E}_0$">|; 

$key = q/{displaymath}T_1,V_b^{,gamma-1}=T_2,V_c^{,gamma-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img900.png"
 ALT="\begin{displaymath}
T_1  V_b^{ \gamma -1} = T_2  V_c^{ \gamma -1}.
\end{displaymath}">|; 

$key = q/{displaymath}E_r=E_r(x_1,cdots,x_n).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="\begin{displaymath}
E_r = E_r(x_1,\cdots, x_n).
\end{displaymath}">|; 

$key = q/h_0=h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img691.png"
 ALT="$h_0 = h$">|; 

$key = q/{displaymath}c_V=frac{1}{nu}left(frac{partialoverline{E}}{partialT}right)_V=3,R{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1183.png"
 ALT="\begin{displaymath}
c_V = \frac{1}{\nu}\left(\frac{\partial\overline{E}}{\partial T}\right)_V = 3 R
\end{displaymath}">|; 

$key = q/{displaymath}dS=frac{dbar{E}+{mathchar'26mskip-12mud}W}{T}=frac{{mathchar'26mskip-12mud}Q}{T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img586.png"
 ALT="\begin{displaymath}
d S = \frac{d\bar{E} +{\mathchar'26\mskip-12mud}W}{T} =\frac{ {\mathchar'26\mskip-12mud}Q}{T}.
\end{displaymath}">|; 

$key = q/{displaymath}sum_rexp(-beta,E_r),E_r^{~2}=-frac{partial}{partialbeta}!left[sum_rtialbeta}right)^2!left[sum_rexp(-betaE_r)right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="622" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img983.png"
 ALT="\begin{displaymath}
\sum_r \exp(-\beta  E_r) E_r^{&nbsp;2} = -\frac{\partial}
{\par...
...rtial \beta}\right)^2\! \left[
\sum_r \exp(-\beta E_r)\right].
\end{displaymath}">|; 

$key = q/J=5slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$J=5/2$">|; 

$key = q/{displaymath}Z^{(0)}=Z,Z',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1015.png"
 ALT="\begin{displaymath}
Z^{(0)} = Z  Z',
\end{displaymath}">|; 

$key = q/{mitOmega}(E)rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="${\mit\Omega}(E)\rightarrow \infty$">|; 

$key = q/{displaymath}bar{n}_s=frac{1}{{{rm{e}^{,beta,epsilon_s}-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1424.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{1}{{\rm e}^{ \beta \epsilon_s}-1}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{f({{bf{v}),d^3{{bf{v}}{n}=left[frac{g(v_x),dv_x}{n}right]left[),dv_y}{n}right]left[frac{g(v_z),dv_z}{n}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1305.png"
 ALT="\begin{displaymath}
\frac{ f({\bf v})   d^3{\bf v}}{n} =\left[\frac{g(v_x) dv_...
...ac{g(v_y) dv_y}{n}\right]\left[\frac{g(v_z) dv_z}{n}\right],
\end{displaymath}">|; 

$key = q/C_VslashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$C_V/T$">|; 

$key = q/2,f=6,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="$2 f=6 N$">|; 

$key = q/{displaymath}sum_rexp(-beta,E_r),E_r=-sum_rfrac{partial}{partialbeta}exp(-beta,E_r)=-frac{partialZ}{partialbeta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img977.png"
 ALT="\begin{displaymath}
\sum_r \exp(-\beta  E_r) E_r = -\sum_r \frac{\partial}{\pa...
...beta}
\exp(-\beta  E_r)
= -\frac{\partial Z}{\partial \beta},
\end{displaymath}">|; 

$key = q/p_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$p_y$">|; 

$key = q/L_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1203.png"
 ALT="$L_y$">|; 

$key = q/{displaymath}2,frac{V}{(2pi)^3}left(frac{4}{3},pi,k_F^{,3}right)=N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1559.png"
 ALT="\begin{displaymath}
2 \frac{V}{(2\pi)^3}\left(\frac{4}{3} \pi k_F^{ 3}\right) = N.
\end{displaymath}">|; 

$key = q/deltap_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img690.png"
 ALT="$\delta p_i$">|; 

$key = q/P,V=nu,R,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img626.png"
 ALT="$P V=\nu  R T$">|; 

$key = q/S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$S$">|; 

$key = q/{displaymath}E=frac{3}{2},N,k,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img780.png"
 ALT="\begin{displaymath}
E = \frac{3}{2}  N  k T.
\end{displaymath}">|; 

$key = q/epsilon>mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1545.png"
 ALT="$\epsilon&gt;\mu$">|; 

$key = q/displaystyled^3{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="$\displaystyle d^3{\bf r}$">|; 

$key = q/displaystylelambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="$\displaystyle \lambda$">|; 

$key = q/,{mathchar'26mskip-12mud}W=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="$ {\mathchar'26\mskip-12mud}W=0$">|; 

$key = q/displaystyle1slash2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1375.png"
 ALT="$\displaystyle 1/2,$">|; 

$key = q/{displaymath}alpha_sequivfrac{partiallnZ_s}{partialN}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1402.png"
 ALT="\begin{displaymath}
\alpha_s\equiv \frac{\partial \ln Z_s}{\partial N}.
\end{displaymath}">|; 

$key = q/(x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="$(x$">|; 

$key = q/{displaymath}lnP(bar{E}_f)>lnP(E_i),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img518.png"
 ALT="\begin{displaymath}
\ln P(\bar{E}_f)&gt; \ln P(E_i),
\end{displaymath}">|; 

$key = q/{displaymath}overline{({mitDelta}n_1)^2}=overline{(n_1)^2}-(overline{n_1})^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="\begin{displaymath}
\overline{({\mit\Delta} n_1)^2}= \overline{(n_1)^2} - (\overline{n_1})^2.
\end{displaymath}">|; 

$key = q/8.5times10^{11};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1169.png"
 ALT="$8.5\times 10^{11}$">|; 

$key = q/bar{E}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="$\bar{E}_i$">|; 

$key = q/{displaymath}chi=frac{N_0,mu^2}{k,T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img966.png"
 ALT="\begin{displaymath}
\chi = \frac{N_0  \mu^2}{k T}.
\end{displaymath}">|; 

$key = q/{displaymath}deltaq,deltap=h_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="\begin{displaymath}
\delta q  \delta p = h_0,
\end{displaymath}">|; 

$key = q/{displaymath}frac{{mitDelta}^astn_1}{overline{n_1}}=frac{sqrt{N,p,q}}{N,p}=sqrt{frac{q}{p}}frac{1}{sqrt{N}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="\begin{displaymath}
\frac{{\mit\Delta}^\ast n_1}{\overline{n_1}}= \frac{\sqrt{N  p q}}{N p} =
\sqrt{\frac{q}{p}}\frac{1}{\sqrt{N}}.
\end{displaymath}">|; 

$key = q/{displaymath}ointdFequiv0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="\begin{displaymath}
\oint dF \equiv 0.
\end{displaymath}">|; 

$key = q/epsilonggmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1537.png"
 ALT="$\epsilon\gg \mu$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W=frac{1}{betaZ}frac{partialZ}{partialx},dx=frac{1}{beta}frac{partiallnZ}{partialx},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img992.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W = \frac{1}{\beta Z}\frac{\partia...
...} dx = \frac{1}{\beta}
\frac{\partial \ln Z}{\partial x} dx.
\end{displaymath}">|; 

$key = q/{displaymath}Hequivoverline{lnP_r}equivsum_rP_rlnP_r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="\begin{displaymath}
H\equiv \overline{\ln P_r} \equiv \sum_r P_r \ln P_r.
\end{displaymath}">|; 

$key = q/lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$\lambda$">|; 

$key = q/H_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="$H_0$">|; 

$key = q/{displaymath}overline{epsilon}_i=frac{1}{2},k,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1082.png"
 ALT="\begin{displaymath}
\overline{\epsilon}_i = \frac{1}{2}  k  T.
\end{displaymath}">|; 

$key = q/{displaymath}overline{E}=hbar,omegaleft[frac{1}{2}+frac{1}{exp(beta,hbar,omega)-1}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1098.png"
 ALT="\begin{displaymath}
\overline{E} = \hbar  \omega \left[ \frac{1}{2} + \frac{1}{\exp(\beta  \hbar 
\omega)-1}
\right].
\end{displaymath}">|; 

$key = q/T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img704.png"
 ALT="$T_0$">|; 

$key = q/{displaymath}p,V=nu,R,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img730.png"
 ALT="\begin{displaymath}
p V = \nu  R T.
\end{displaymath}">|; 

$key = q/{displaymath}Sequivk,(lnZ+beta,overline{E}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1006.png"
 ALT="\begin{displaymath}
S \equiv k (\ln Z +\beta \overline{E}).
\end{displaymath}">|; 

$key = q/displaystylesigma_D(omega)=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1235.png"
 ALT="$\displaystyle \sigma_D(\omega)=$">|; 

$key = q/{displaymath}bar{p}=-frac{partialbar{E}}{partialV},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="\begin{displaymath}
\bar{p} = - \frac{ \partial\bar{E} }{\partial V},
\end{displaymath}">|; 

$key = q/{displaymath}c=frac{1}{nu}frac{{mathchar'26mskip-12mud}Q}{dT}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img754.png"
 ALT="\begin{displaymath}
c = \frac{1}{\nu}\frac{{\mathchar'26\mskip-12mud}Q}{dT}.
\end{displaymath}">|; 

$key = q/tilde{x}_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img570.png"
 ALT="$\tilde{x}_\alpha$">|; 

$key = q/5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1130.png"
 ALT="$5$">|; 

$key = q/{displaymath}P(y)propto{mitOmega}(y).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img610.png"
 ALT="\begin{displaymath}
P(y) \propto {\mit\Omega}(y).
\end{displaymath}">|; 

$key = q/{displaymath}Ksimeqfrac{3}{4},frac{V,c}{{mitLambda}^3}left[p_F^{,4}+m^2,c^2,p_F^{,2}+cdotsright].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1624.png"
 ALT="\begin{displaymath}
K \simeq \frac{3}{4} \frac{V c}{{\mit\Lambda}^3}
\left[p_F^{ 4} + m^2 c^2 p_F^{ 2}+\cdots \right].
\end{displaymath}">|; 

$key = q/{displaymath}overline{f(u)+g(u)}=overline{f(u)}+overline{g(u)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="\begin{displaymath}
\overline{f(u)+g(u)}= \overline{f(u)}+\overline{g(u)}.
\end{displaymath}">|; 

$key = q/q_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="$q_x$">|; 

$key = q/f=3,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="$f=3 N$">|; 

$key = q/{displaymath}{mitDelta}bar{E}=Q,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="\begin{displaymath}
{\mit\Delta} \bar{E} = Q,
\end{displaymath}">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$u$">|; 

$key = q/{displaymath}E=frac{1}{2,m}sum_{i=1}^Nsum_{alpha=1}^3p_{i,alpha}^{~~2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="\begin{displaymath}
E = \frac{1}{2 m} \sum_{i=1}^N \sum_{\alpha=1}^3 p_{i \alpha}^{&nbsp;&nbsp;2},
\end{displaymath}">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W=bar{p},,dV,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W = \bar{p}   dV,
\end{displaymath}">|; 

$key = q/{displaymath}Z=sum_rexp(-beta,E_r).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img978.png"
 ALT="\begin{displaymath}
Z = \sum_r \exp(-\beta  E_r).
\end{displaymath}">|; 

$key = q/q_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$q_i$">|; 

$key = q/{displaymath}S(E_0)=kln1=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img696.png"
 ALT="\begin{displaymath}
S(E_0) = k\ln 1 = 0.
\end{displaymath}">|; 

$key = q/{displaymath}overline{v_z}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1302.png"
 ALT="\begin{displaymath}
\overline{v_z} = 0,
\end{displaymath}">|; 

$key = q/{displaymath}F(epsilon)=frac{1}{{{rm{e}^{,beta,(epsilon-mu)}+1}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1528.png"
 ALT="\begin{displaymath}
F(\epsilon) = \frac{1}{{\rm e}^{ \beta (\epsilon-\mu)} + 1}
\end{displaymath}">|; 

$key = q/ln{calP};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$\ln{\cal P}$">|; 

$key = q/rho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img861.png"
 ALT="$\rho_0$">|; 

$key = q/5slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img974.png"
 ALT="$5/2$">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img526.png"
 ALT="$k$">|; 

$key = q/{displaymath}S=k,ln{mitOmega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img627.png"
 ALT="\begin{displaymath}
S = k  \ln{\mit\Omega}.
\end{displaymath}">|; 

$key = q/f({{bf{v})equivf(v);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1318.png"
 ALT="$f({\bf v})\equiv f(v)$">|; 

$key = q/{displaymath}c_V^{(e)}=gamma,T,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1584.png"
 ALT="\begin{displaymath}
c_V^{(e)} = \gamma T,
\end{displaymath}">|; 

$key = q/P(XmidY);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$P(X\mid Y)$">|; 

$key = q/{{bf{r_i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="${\bf r_i}$">|; 

$key = q/{displaymath}lnZ_{{rm{MB}=N,ln!left(sum_r{{rm{e}^{-beta,epsilon_r}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1457.png"
 ALT="\begin{displaymath}
\ln Z_{\rm MB} = N \ln\!\left(\sum_r {\rm e}^{-\beta \epsilon_r}\right).
\end{displaymath}">|; 

$key = q/z_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="$z_i)$">|; 

$key = q/lambdaequiv2pislashk>lambda_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1564.png"
 ALT="$\lambda\equiv 2\pi/k &gt; \lambda_F$">|; 

$key = q/{displaymath}E=K+U,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1592.png"
 ALT="\begin{displaymath}
E = K + U,
\end{displaymath}">|; 

$key = q/{displaymath}z_0=frac{R,T}{mu,g}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img819.png"
 ALT="\begin{displaymath}
z_0 = \frac{R T}{\mu  g}
\end{displaymath}">|; 

$key = q/hbar,omegasimeq3,k,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1489.png"
 ALT="$\hbar \omega\simeq 3  k T$">|; 

$key = q/{displaymath}W_{if}=int_i^f{mathchar'26mskip-12mud}W,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="\begin{displaymath}
W_{if} = \int_i^f {\mathchar'26\mskip-12mud}W,
\end{displaymath}">|; 

$key = q/g(v_z),dv_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1293.png"
 ALT="$g(v_z) dv_z$">|; 

$key = q/{mitOmega}_f>{mitOmega}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img615.png"
 ALT="${\mit\Omega}_f &gt;
{\mit\Omega}_i$">|; 

$key = q/displaystylenleft(frac{m}{2pi,k,T}right)^{3slash2}exp(-m,v_z^{~2}slash2,k,T)left(sqrt{frac{2pi,k,T}{m}}right)^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1299.png"
 ALT="$\displaystyle n \left(\frac{m}{2\pi  k T}\right)^{3/2}\exp(-m v_z^{&nbsp;2}/ 2 k T)
\left(\sqrt{\frac{2\pi k T}{m}}\right)^2,$">|; 

$key = q/sigma(x,y)=c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="$\sigma (x, y) = c$">|; 

$key = q/dV+dV'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img675.png"
 ALT="$dV + dV' =0$">|; 

$key = q/{{rm{m}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img818.png"
 ALT="${\rm m}^{-2}$">|; 

$key = q/{displaymath}P(XmidY)=P(X)+P(Y).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="\begin{displaymath}
P(X\mid Y) = P(X) + P(Y).
\end{displaymath}">|; 

$key = q/{displaymath}frac{{mathchar'26mskip-12mud}G}{tau}=dsigma.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="\begin{displaymath}
\frac{{\mathchar'26\mskip-12mud}G}{\tau} = d\sigma.
\end{displaymath}">|; 

$key = q/E+deltaE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="$E+\delta E$">|; 

$key = q/{{rm{N}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img773.png"
 ALT="${\rm N}_2$">|; 

$key = q/273.16^circ,{{rm{K};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img531.png"
 ALT="$273.16^\circ {\rm K}$">|; 

$key = q/{displaymath}S=klnleft(intcdotsintdq_1cdotsdq_f,dp_1cdotsdp_fright)-kflnh_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="412" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img687.png"
 ALT="\begin{displaymath}
S = k \ln\left( \int \cdots \int dq_1\cdots dq_f dp_1\cdots dp_f\right) -
kf\ln h_0.
\end{displaymath}">|; 

$key = q/{displaymath}q_1=w+q_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img879.png"
 ALT="\begin{displaymath}
q_1 = w + q_2.
\end{displaymath}">|; 

$key = q/lnP_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$\ln P_r$">|; 

$key = q/p_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1065.png"
 ALT="$p_k$">|; 

$key = q/{mitOmega}_fgg{mitOmega}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img599.png"
 ALT="${\mit\Omega}_f\gg {\mit\Omega}_i$">|; 

$key = q/{displaymath}Z=sum_{R}{{rm{e}^{-beta,(n_1,epsilon_1+n_2,epsilon_2+cdots)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1435.png"
 ALT="\begin{displaymath}
Z=\sum_{R} {\rm e}^{-\beta (n_1 \epsilon_1+n_2 \epsilon_2+\cdots)},
\end{displaymath}">|; 

$key = q/{displaymath}P(E)=P(tilde{E})expleft[-frac{1}{2},lambda_0,(E-tilde{E})^2right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="\begin{displaymath}
P(E)=P(\tilde{E}) \exp\left[-\frac{1}{2} \lambda_0  (E-\tilde{E})^2\right],
\end{displaymath}">|; 

$key = q/p_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="$p_z$">|; 

$key = q/W_{rs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="$W_{rs}$">|; 

$key = q/n=overline{n_1}pm{mitDelta}^astn_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="$n=\overline{n_1}\pm {\mit\Delta}^\ast n_1$">|; 

$key = q/dbar{E}=bar{E}_f-bar{E}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="$d\bar{E} = \bar{E}_f - \bar{E}_i$">|; 

$key = q/{{rm{O}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img774.png"
 ALT="${\rm O}_2$">|; 

$key = q/dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img663.png"
 ALT="$dV$">|; 

$key = q/3N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="$3N$">|; 

$key = q/{displaymath}frac{w}{T_2}leqq_1left(frac{1}{T_2}-frac{1}{T_1}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img883.png"
 ALT="\begin{displaymath}
\frac{w}{T_2} \leq q_1\left(\frac{1}{T_2} - \frac{1}{T_1}\right).
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}(E,V)proptoint^{E+deltaE}_Ed^3{{bf{r}_1cdotsd^3{{bf{r}_N,d^3{{bf{p_1}cdotsd^3{{bf{p_N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="\begin{displaymath}
{\mit\Omega}(E, V) \propto \int^{E+\delta E}_Ed^3{\bf r}_1\cdots d^3{\bf r}_N 
d^3 {\bf p_1}\cdots d^3 {\bf p_N}.
\end{displaymath}">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W=sum_alphabar{X}_alpha,dbar{x}_alpha.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img585.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W = \sum_\alpha \bar{X}_\alpha d\bar{x}_\alpha.
\end{displaymath}">|; 

$key = q/n_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1381.png"
 ALT="$n_r$">|; 

$key = q/{displaymath}P_3(2)=p,p,q+q,p,p+p,q,p=3,p^2,q{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="\begin{displaymath}
P_3(2) = p p q + q p p + p q p = 3  p^2 q
\end{displaymath}">|; 

$key = q/{displaymath}rho,d^3{{bf{k}=left(frac{L_x}{2pi},dk_xright)left(frac{L_y}{2pi},dk2pi},dk_zright)=frac{V}{(2pi)^3},dk_x,dk_y,dk_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="477" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1222.png"
 ALT="\begin{displaymath}
\rho  d^3{\bf k} = \left(\frac{L_x}{2\pi} dk_x\right)
\le...
...{2\pi} dk_z\right)=
\frac{V}{(2\pi)^3}  dk_x  dk_y  dk_z,
\end{displaymath}">|; 

$key = q/F(epsilon)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1536.png"
 ALT="$F(\epsilon)=1$">|; 

$key = q/displaystyled^3{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="$\displaystyle d^3{\bf p}$">|; 

$key = q/p_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="$p_x$">|; 

$key = q/displaystylenleft(frac{m}{2pi,k,T}right)^{3slash2}exp(-m,v_z^{~2}slash2,k,T)left[int_{-infty}^{infty}exp(-m,v_x^{~2}slash2,k,T)right]^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="486" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1298.png"
 ALT="$\displaystyle n \left(\frac{m}{2\pi  k T}\right)^{3/2} \exp(-m v_z^{&nbsp;2}/ 2 k T)
\left[\int_{-\infty}^{\infty} \exp(-m v_x^{&nbsp;2}/ 2 k T)\right]^2$">|; 

$key = q/{displaymath}V=frac{4pi}{3},R^3{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1599.png"
 ALT="\begin{displaymath}
V = \frac{4\pi}{3} R^3
\end{displaymath}">|; 

$key = q/dbar{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="$d\bar{p}$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Qequivdbar{E}+{mathchar'26mskip-12mud}W.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q \equiv d\bar{E} + {\mathchar'26\mskip-12mud}W.
\end{displaymath}">|; 

$key = q/C=(NslashV)(beta,mslash2pi)^{3slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1289.png"
 ALT="$ C =(N/V) (\beta  m / 2\pi)^{3/2}$">|; 

$key = q/b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img895.png"
 ALT="$b$">|; 

$key = q/{displaymath}B_2=-frac{1}{({mitDelta}^astn_1)^2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="\begin{displaymath}
B_2 = -\frac{1}{({\mit\Delta}^\ast n_1)^2}
\end{displaymath}">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$x$">|; 

$key = q/overline{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img999.png"
 ALT="$\overline{p}$">|; 

$key = q/c=3times10^8,{{rm{m},{{rm{s}^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1462.png"
 ALT="$c=3\times 10^8 {\rm m} {\rm s}^{-1}$">|; 

$key = q/3,p^2,q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$3 p^2 q$">|; 

$key = q/displaystyleT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img678.png"
 ALT="$\displaystyle T$">|; 

$key = q/{displaymath}w+q_2=q_1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img912.png"
 ALT="\begin{displaymath}
w + q_2 = q_1.
\end{displaymath}">|; 

$key = q/(2pi)^{-3},V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1555.png"
 ALT="$(2\pi)^{-3} V$">|; 

$key = q/{displaymath}frac{dsigma}{dx}equivfrac{partialsigma}{partialx}+frac{partialsigma}{partialy}frac{dy}{dx}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="\begin{displaymath}
\frac{d\sigma}{d x}\equiv \frac{\partial \sigma}{\partial x}
+\frac{\partial\sigma}{\partial y}\frac{dy}{dx} = 0.
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}(E,V)=B,V^NE^{,3Nslash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="\begin{displaymath}
{\mit\Omega}(E, V) = B V^N E^{ 3N/2},
\end{displaymath}">|; 

$key = q/5000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1132.png"
 ALT="$5000$">|; 

$key = q/lambda_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="$\lambda_0&gt;0$">|; 

$key = q/{displaymath}P(n,n+dn)={calP}(n),dn,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="\begin{displaymath}
P(n, n+dn) = {\cal P}(n)  dn,
\end{displaymath}">|; 

$key = q/k,Tggmu,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img959.png"
 ALT="$k  T\gg \mu  B$">|; 

$key = q/c_V,p,V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img793.png"
 ALT="$c_V  p  V$">|; 

$key = q/{displaymath}S'=2,nu,Rleft[ln2,V+frac{3}{2}lnT+sigmaright]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1047.png"
 ALT="\begin{displaymath}
S' = 2 \nu  R \left[\ln 2 V + \frac{3}{2} \ln T + \sigma\right]
\end{displaymath}">|; 

$key = q/T';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img657.png"
 ALT="$T'$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{fermi.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="522" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img1541.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{fermi.eps}}
\end{figure}">|; 

$key = q/F=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1544.png"
 ALT="$F=0$">|; 

$key = q/{displaymath}P_r=C',{mitOmega}'(E^{(0)}-E_r),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img922.png"
 ALT="\begin{displaymath}
P_r = C'  {\mit\Omega}'(E^{(0)} - E_r),
\end{displaymath}">|; 

$key = q/{displaymath}dlnZ=frac{partiallnZ}{partialx},dx+frac{partiallnZ}{partialbeta},dbeta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1001.png"
 ALT="\begin{displaymath}
d \ln Z = \frac{\partial \ln Z}{\partial x} dx + \frac{\partial \ln Z}
{\partial \beta} d\beta.
\end{displaymath}">|; 

$key = q/p=n,k,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1337.png"
 ALT="$p = n k T$">|; 

$key = q/{displaymath}{mitOmega}(XotimesY)={mitOmega}(X),{mitOmega}(Y).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="\begin{displaymath}
{\mit\Omega}(X\otimes Y) = {\mit\Omega}(X)  {\mit\Omega}(Y).
\end{displaymath}">|; 

$key = q/bar{E}+dbar{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img574.png"
 ALT="$\bar{E}+d\bar{E}$">|; 

$key = q/8.85;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img833.png"
 ALT="$8.85$">|; 

$key = q/Trightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1043.png"
 ALT="$T\rightarrow 0$">|; 

$key = q/F(v),dv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1309.png"
 ALT="$F(v)  dv$">|; 

$key = q/dE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img666.png"
 ALT="$dE$">|; 

$key = q/{displaymath}S=nu,Rleft[lnV+frac{3}{2}lnT+sigmaright]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1046.png"
 ALT="\begin{displaymath}
S = \nu  R \left[\ln V + \frac{3}{2} \ln T + \sigma\right]
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}^astn_1=sqrt{N,p,q}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="\begin{displaymath}
{\mit\Delta}^\ast n_1 = \sqrt{N p q}.
\end{displaymath}">|; 

$key = q/{displaymath}ln{mitOmega}_f-ln{mitOmega}_iequiv{mitDelta}ln{mitOmega}geq0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img620.png"
 ALT="\begin{displaymath}
\ln {\mit\Omega}_f -\ln {\mit\Omega}_i \equiv {\mit\Delta} \ln{\mit\Omega}\geq 0
\end{displaymath}">|; 

$key = q/dE+dE'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img674.png"
 ALT="$ dE + dE' = 0$">|; 

$key = q/{displaymath}frac{1}{k,T}equivbetaequivfrac{partialln{mitOmega}}{partialE},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img525.png"
 ALT="\begin{displaymath}
\frac{1}{k T} \equiv \beta \equiv \frac{\partial \ln{\mit\Omega}}{\partial E},
\end{displaymath}">|; 

$key = q/displaystyleq.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$\displaystyle q.$">|; 

$key = q/,{mathchar'26mskip-12mud}Q'=-,{mathchar'26mskip-12mud}Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img654.png"
 ALT="$ {\mathchar'26\mskip-12mud}Q' = -  {\mathchar'26\mskip-12mud}Q$">|; 

$key = q/j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1347.png"
 ALT="$j$">|; 

$key = q/{displaymath}bar{n}_s=frac{1}{beta}frac{partial}{partialepsilon_s}ln,(1-{{rm{e}^{-beta,epsilon_s}}{1-{{rm{e}^{-beta,epsilon_s}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1423.png"
 ALT="\begin{displaymath}
\bar{n}_s =\frac{1}{\beta} \frac{\partial}{\partial\epsilon_...
...\rm e}^{-\beta \epsilon_s}}
{1-{\rm e}^{-\beta \epsilon_s}},
\end{displaymath}">|; 

$key = q/{displaymath}P({{bf{r},{{bf{p}),d^3{{bf{r},d^3{{bf{p}proptoexp(-beta,p^2slash2m),d^3{{bf{r},d^3{{bf{p}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1277.png"
 ALT="\begin{displaymath}
P({\bf r}, {\bf p})  d^3{\bf r} d^3{\bf p} \propto \exp(-\beta  p^2/2m)
 d^3{\bf r} d^3{\bf p}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}bar{E}=bar{E}_f-bar{E}_i=int_{i}^fdbar{E}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="\begin{displaymath}
{\mit\Delta} \bar{E} = \bar{E}_f - \bar{E}_i = \int_{i}^f d\bar{E}.
\end{displaymath}">|; 

$key = q/{displaymath}tanhysimeq1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img958.png"
 ALT="\begin{displaymath}
\tanh y \simeq 1.
\end{displaymath}">|; 

$key = q/{displaymath}(rho,T)_{{rm{packet}=(rho,T)_{{rm{atmosphere}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img847.png"
 ALT="\begin{displaymath}
(\rho  T)_{\rm packet} = (\rho  T)_{\rm atmosphere}.
\end{displaymath}">|; 

$key = q/A';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="$A'$">|; 

$key = q/p(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img803.png"
 ALT="$p(z)$">|; 

$key = q/bar{n}_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1432.png"
 ALT="$\bar{n}_s$">|; 

$key = q/displaystylek_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1213.png"
 ALT="$\displaystyle k_z$">|; 

$key = q/u_2,cdots,u_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$u_2, \cdots, u_M$">|; 

$key = q/{displaymath}T,dS=dE+p,dV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img717.png"
 ALT="\begin{displaymath}
T dS = dE + p dV.
\end{displaymath}">|; 

$key = q/E_{s}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1007.png"
 ALT="$E_{s}'$">|; 

$key = q/E+delta^astE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img632.png"
 ALT="$E+\delta^\ast E$">|; 

$key = q/e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1583.png"
 ALT="$e$">|; 

$key = q/{displaymath}P_{{rm{tot}(T)=frac{k^4,T^4}{4pi^2,c^2,hbar^3}int_0^inftyfrac{eta^3,deta}{expeta-1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1504.png"
 ALT="\begin{displaymath}
P_{\rm tot}(T) = \frac{k^4  T^4 }{4\pi^2  c^2 \hbar^3} \int_0^\infty
\frac{\eta^3 d\eta}{\exp\eta -1},
\end{displaymath}">|; 

$key = q/k,N_A=R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img729.png"
 ALT="$k N_A = R$">|; 

$key = q/24.9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1184.png"
 ALT="$24.9$">|; 

$key = q/displaystyleE(T,V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1108.png"
 ALT="$\displaystyle E (T, V)$">|; 

$key = q/{displaymath}B_2=-frac{1}{tilde{n}}-frac{1}{N-tilde{n}}=-frac{1}{Np}-frac{1}{N,(1-p)}=-frac{1}{N,p,q},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="\begin{displaymath}
B_2 = -\frac{1}{\tilde{n}}-\frac{1}{N-\tilde{n}} =
-\frac{1}{Np}-\frac{1}{N (1-p)}= - \frac{1}{N p q},
\end{displaymath}">|; 

$key = q/h_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="$h_2$">|; 

$key = q/{displaymath}tanhyequivfrac{exp(y)-exp(-y)}{exp(y)+exp(-y)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img956.png"
 ALT="\begin{displaymath}
\tanh y \equiv \frac{ \exp(y) - \exp(-y)}{\exp(y)+\exp(-y)}.
\end{displaymath}">|; 

$key = q/hbar,omegaggk,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1485.png"
 ALT="$\hbar \omega\gg k T$">|; 

$key = q/{displaymath}U=-frac{3}{5}frac{G,M^2}{R},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1595.png"
 ALT="\begin{displaymath}
U = -\frac{3}{5}\frac{G M^2}{R},
\end{displaymath}">|; 

$key = q/omega+domega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1227.png"
 ALT="$\omega+d \omega$">|; 

$key = q/Q_i+dQ_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1345.png"
 ALT="$Q_i+dQ_i$">|; 

$key = q/fsimN_Asimeq10^{24};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="$f\sim N_A\simeq 10^{24}$">|; 

$key = q/ln{calP}(n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$\ln{\cal P}(n)$">|; 

$key = q/mid;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$\mid$">|; 

$key = q/N_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img963.png"
 ALT="$N_0$">|; 

$key = q/displaystyles_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1351.png"
 ALT="$\displaystyle s_i$">|; 

$key = q/{displaymath}E_requivE_r(x_1,x_2,cdots,x_n).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="\begin{displaymath}
E_r \equiv E_r(x_1, x_2, \cdots, x_n).
\end{displaymath}">|; 

$key = q/{displaymath}|{mitPsi}(cdotsQ_icdotsQ_jcdots)|^2=|{mitPsi}(cdotsQ_jcdotsQ_icdots)|^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1354.png"
 ALT="\begin{displaymath}
\vert{\mit\Psi}(\cdots Q_i\cdots Q_j\cdots)\vert^2=
\vert{\mit\Psi}(\cdots Q_j\cdots Q_i\cdots)\vert^2.
\end{displaymath}">|; 

$key = q/displaystylefrac{int_{-infty}^{infty}exp(-beta,epsilon_i),epsilon_i,dp_i,int_{-ilon_i),dp_i,int_{-infty}^{infty}exp(-beta,E'),dq_1cdotsdp_f},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1074.png"
 ALT="$\displaystyle \frac{\int_{-\infty}^{\infty} \exp(-\beta  \epsilon_i) \epsilon...
...psilon_i)  dp_i
 \int_{-\infty}^{\infty} \exp(-\beta  E') dq_1\cdots dp_f},$">|; 

$key = q/3.47;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1168.png"
 ALT="$3.47$">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1596.png"
 ALT="$G$">|; 

$key = q/intdbar{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="$\int d\bar{E}$">|; 

$key = q/E'=E^{(0)}-E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="$E' = E^{(0)}-E$">|; 

$key = q/lambda(tilde{E})>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="$\lambda(\tilde{E})&gt;0$">|; 

$key = q/{displaymath}frac{N!}{n_1!,n_2!,cdots}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1436.png"
 ALT="\begin{displaymath}
\frac{N!}{n_1 ! n_2! \cdots}
\end{displaymath}">|; 

$key = q/dS'=,{mathchar'26mskip-12mud}Q'slashT';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img656.png"
 ALT="$dS' =  {\mathchar'26\mskip-12mud}Q'/T'$">|; 

$key = q/displaystyleleft(frac{partialS}{partialT}right)_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img742.png"
 ALT="$\displaystyle \left(\frac{\partial S}{\partial T}\right)_V$">|; 

$key = q/{mitPsi}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="${\mit\Psi}_r$">|; 

$key = q/{displaymath}left(frac{partialE}{partialV}right)_T=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img749.png"
 ALT="\begin{displaymath}
\left(\frac{\partial E}{\partial V}\right)_T =0,
\end{displaymath}">|; 

$key = q/V_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="$V_i$">|; 

$key = q/{displaymath}S_f-S_i=nu,R,ln2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img628.png"
 ALT="\begin{displaymath}
S_f - S_i = \nu  R  \ln 2,
\end{displaymath}">|; 

$key = q/{displaymath}n_1,p^{n_1}equivp,frac{partial}{partialp},p^{n_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="\begin{displaymath}
n_1 p^{n_1} \equiv p \frac{\partial}{\partial p} p^{n_1},
\end{displaymath}">|; 

$key = q/Srightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1185.png"
 ALT="$S\rightarrow \infty$">|; 

$key = q/{displaymath}(c_V)_{{rm{translation}=frac{3}{2},R.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1145.png"
 ALT="\begin{displaymath}
(c_V)_{\rm translation} = \frac{3}{2} R.
\end{displaymath}">|; 

$key = q/{displaymath}dS=frac{1}{T}left(frac{partialE}{partialT}right)_VdT+left[frac{1}{TrtialE}{partialV}right)_T+frac{nu,R}{V}right]dV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img738.png"
 ALT="\begin{displaymath}
dS = \frac{1}{T} \left(\frac{\partial E}{\partial T}\right)_...
...partial E}{\partial V}\right)_T +
\frac{\nu  R}{V}\right] dV.
\end{displaymath}">|; 

$key = q/{mitDelta}NllN;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1399.png"
 ALT="${\mit\Delta}N\ll N$">|; 

$key = q/displaystyleB_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$\displaystyle B_1$">|; 

$key = q/A_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="$A_2$">|; 

$key = q/f(v);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1319.png"
 ALT="$f(v)$">|; 

$key = q/{displaymath}overline{(u-bar{u})^2}=overline{u^2}-bar{u}^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="\begin{displaymath}
\overline{(u-\bar{u})^2}= \overline{u^2}-\bar{u}^2.
\end{displaymath}">|; 

$key = q/,{mathchar'26mskip-12mud}W=p,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img889.png"
 ALT="$ {\mathchar'26\mskip-12mud}W = p  dV$">|; 

$key = q/{displaymath}dE_r=sum_{alpha=1}^nfrac{partialE_r}{partialx_alpha},dx_alpha.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="\begin{displaymath}
d E_r = \sum_{\alpha =1}^n \frac{\partial E_r}{\partial x_\alpha} dx_\alpha.
\end{displaymath}">|; 

$key = q/w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img868.png"
 ALT="$w$">|; 

$key = q/k_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1207.png"
 ALT="$k_y$">|; 

$key = q/3,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$3 N$">|; 

$key = q/tilde{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img612.png"
 ALT="$\tilde{y}$">|; 

$key = q/rho_{{rm{packet}>rho_{{rm{atmosphere};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img849.png"
 ALT="$\rho_{\rm packet} &gt;
\rho_{\rm atmosphere}$">|; 

$key = q/[p(z)-p(z+dz)]A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img805.png"
 ALT="$[p(z) - p(z+dz)]A$">|; 

$key = q/{displaymath}q_1=int_a^bp,dV,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img893.png"
 ALT="\begin{displaymath}
q_1 = \int_a^b p  dV,
\end{displaymath}">|; 

$key = q/p_F=hbar,k_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1551.png"
 ALT="$p_F=\hbar k_F$">|; 

$key = q/{displaymath}frac{-q_1}{T_1}+frac{q_1-w}{T_2}geq0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img882.png"
 ALT="\begin{displaymath}
\frac{-q_1}{T_1} + \frac{q_1 - w}{T_2} \geq 0,
\end{displaymath}">|; 

$key = q/{displaymath}overline{mu}_B=mutanhfrac{mu,B}{k,T},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img955.png"
 ALT="\begin{displaymath}
\overline{\mu}_B = \mu \tanh\frac{\mu  B}{k T},
\end{displaymath}">|; 

$key = q/{mitOmega}(XmidY);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="${\mit\Omega}(X \mid Y)$">|; 

$key = q/X=partialFslashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$X= \partial F/\partial x$">|; 

$key = q/{displaymath}M=2,N,m_p,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1606.png"
 ALT="\begin{displaymath}
M = 2 N m_p,
\end{displaymath}">|; 

$key = q/k_y+dk_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1220.png"
 ALT="$k_y + d k_y $">|; 

$key = q/R(E)=(2,m,E)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="$R(E) = (2 m  E)^{1/2}$">|; 

$key = q/beta_i'equivbeta'(E_i');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img523.png"
 ALT="$\beta_i'\equiv\beta'(E_i')$">|; 

$key = q/(4slash3),pi,k_F^{,3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1557.png"
 ALT="$(4/3) \pi k_F^{ 3}$">|; 

$key = q/{displaymath}overline{E}=3,N,k,T=3,nu,R,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1182.png"
 ALT="\begin{displaymath}
\overline{E} = 3  N k T = 3 \nu  R T.
\end{displaymath}">|; 

$key = q/q=1-p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$q=1-p$">|; 

$key = q/{displaymath}frac{q_1}{T_1}+frac{-q_2}{T_2}geq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img913.png"
 ALT="\begin{displaymath}
\frac{q_1}{T_1} + \frac{-q_2}{T_2} \geq 0.
\end{displaymath}">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="$z$">|; 

$key = q/{displaymath}overline{(u-bar{u})^2}=overline{(u^2-2,u,bar{u}+bar{u}^2)}=overline{u^2}-2,bar{u},bar{u}+bar{u}^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="\begin{displaymath}
\overline{(u-\bar{u})^2} = \overline{(
u^2-2 u \bar{u}+\bar{u}^2)}= \overline{u^2}-2 \bar{u} \bar{u}+\bar{u}^2,
\end{displaymath}">|; 

$key = q/{displaymath}frac{dp}{dz}=-rho,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img808.png"
 ALT="\begin{displaymath}
\frac{dp}{d z} = - \rho  g.
\end{displaymath}">|; 

$key = q/N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$N$">|; 

$key = q/{displaymath}frac{partial^2S}{partialVpartialT}=frac{partial^2S}{partialTpartialV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img746.png"
 ALT="\begin{displaymath}
\frac{\partial ^2 S}{\partial V \partial T} = \frac{\partial ^2 S}
{\partial T \partial V}.
\end{displaymath}">|; 

$key = q/{displaymath}rho_k(k),dk=frac{4pi,k^2,dk}{(2pi)^3}=frac{k^2}{2pi^2},dk.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1472.png"
 ALT="\begin{displaymath}
\rho_k(k) dk = \frac{4\pi  k^2 dk}{(2\pi)^3} = \frac{k^2}{2\pi^2}  dk.
\end{displaymath}">|; 

$key = q/v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1323.png"
 ALT="$v^2$">|; 

$key = q/9.8^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img842.png"
 ALT="$9.8^\circ$">|; 

$key = q/{displaymath}sum_rfrac{1}{{{rm{e}^{,alpha+beta,epsilon_r}+1}=N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1412.png"
 ALT="\begin{displaymath}
\sum_r \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_r}+ 1} = N.
\end{displaymath}">|; 

$key = q/n_r=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1391.png"
 ALT="$n_r=0$">|; 

$key = q/0,1,2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1364.png"
 ALT="$0, 1, 2$">|; 

$key = q/v=|{{bf{v}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1310.png"
 ALT="$v=\vert{\bf v}\vert$">|; 

$key = q/displaystylek_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1211.png"
 ALT="$\displaystyle k_y$">|; 

$key = q/omegasimsqrt{kappaslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1180.png"
 ALT="$\omega\sim \sqrt{\kappa/m}$">|; 

$key = q/{displaymath}overline{epsilon}_i=frac{int_{-infty}^{infty}exp(-beta,epsilon_i),e{int_{-infty}^{infty}exp(-beta,epsilon_i),dp_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1075.png"
 ALT="\begin{displaymath}
\overline{\epsilon}_i = \frac{\int_{-\infty}^{\infty} \exp(-...
..._i}
{\int_{-\infty}^{\infty}\exp(-\beta  \epsilon_i)  dp_i}.
\end{displaymath}">|; 

$key = q/lnn!;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$\ln n!$">|; 

$key = q/L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="$L$">|; 

$key = q/{displaymath}X'=tau,frac{partialsigma}{partialx},~~~~~Y'=tau,frac{partialsigma}{partialy}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="\begin{displaymath}
X' = \tau  \frac{\partial \sigma}{\partial x},&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Y' = \tau 
\frac{\partial \sigma}
{\partial y}.
\end{displaymath}">|; 

$key = q/displaystylefrac{2pi}{L_y},n_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1212.png"
 ALT="$\displaystyle \frac{2\pi}{L_y}  n_y,$">|; 

$key = q/9.81,{{rm{m},{{rm{s}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img820.png"
 ALT="$9.81 {\rm m} {\rm s}^{-2}$">|; 

$key = q/{displaymath}int_0^pin(omega,theta),domega,dtheta=n(omega),domega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img1495.png"
 ALT="\begin{displaymath}
\int_0^\pi n(\omega, \theta) d\omega d\theta = n(\omega)
 d\omega.
\end{displaymath}">|; 

$key = q/rho=n,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1338.png"
 ALT="$\rho = n  m$">|; 

$key = q/overline{mu}_Bsimeqmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img953.png"
 ALT="$\overline{\mu}_B\simeq \mu$">|; 

$key = q/displaystyle-frac{partial^2ln{mitOmega}}{partialE^2}=-frac{partialbeta}{partialE}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="$\displaystyle -\frac{\partial ^2 \ln {\mit\Omega}}{\partial E^2}
= - \frac{\partial\beta}
{\partial E}.$">|; 

$key = q/{displaymath}sigma(E,x)=frac{{mitOmega}(E,x)}{deltaE},overline{frac{partialE_r}{partialx}},,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img543.png"
 ALT="\begin{displaymath}
\sigma (E,x) = \frac{{\mit\Omega}(E,x)}{\delta E} 
\overline{\frac{\partial E_r}{\partial x}} 
 dx,
\end{displaymath}">|; 

$key = q/{displaymath}e_i=frac{hbar^2pi^2}{2,m,L^2}left(n_{i1}^{~2}+n_{i2}^{~2}+n_{i3}^{~2}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="\begin{displaymath}
e_i = \frac{\hbar^2 \pi^2}{2 m  L^2}\left( n_{i1}^{&nbsp;2}+n_{i2}^{&nbsp;2}
+n_{i3}^{&nbsp;2}\right),
\end{displaymath}">|; 

$key = q/displaystyleA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1609.png"
 ALT="$\displaystyle A$">|; 

$key = q/J=3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$J=3/2$">|; 

$key = q/{displaymath}Z=sum_{n=0}^inftyexp(-beta,E_n)=exp[-(1slash2),beta,hbar,omega]sum_{n=0}^inftyexp(-n,beta,hbar,omega).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="477" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1092.png"
 ALT="\begin{displaymath}
Z = \sum_{n=0}^\infty \exp(-\beta  E_n) = \exp[-(1/2) \bet...
...  \omega]
\sum_{n=0}^\infty \exp(- n \beta  \hbar \omega).
\end{displaymath}">|; 

$key = q/P_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="$P_r$">|; 

$key = q/N_A=6times10^{23};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img622.png"
 ALT="$N_A=6\times 10^{23}$">|; 

$key = q/0.6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img828.png"
 ALT="$0.6$">|; 

$key = q/displaystylebar{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1572.png"
 ALT="$\displaystyle \bar{E}$">|; 

$key = q/{displaymath}overline{E}=int_0^inftysigma_D(omega),hbar,omegaleft(frac{1}{2}+frac{1}{exp(beta,hbar,omega)-1}right),domega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1246.png"
 ALT="\begin{displaymath}
\overline{E} = \int_0^\infty \sigma_D(\omega)  \hbar \omeg...
...{2} + \frac{1}{\exp(\beta \hbar  \omega)-1}\right) d\omega.
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialln{mitOmega}}{partialx}=frac{partialln{mitOmega}}{partialE},bar{X}+frac{partialbar{X}}{partialE}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img551.png"
 ALT="\begin{displaymath}
\frac{\partial \ln {\mit\Omega}}{\partial x} = \frac{\partia...
...}}{\partial E} 
\bar{X} +\frac{\partial \bar{X}}{\partial E}.
\end{displaymath}">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="$\theta$">|; 

$key = q/{displaymath}frac{partial{mitOmega}(E,x)}{partialx},dx=sigma(E)-sigma(E+deltaE)simeq-frac{partialsigma}{partialE},deltaE,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img548.png"
 ALT="\begin{displaymath}
\frac{\partial {\mit\Omega}(E,x)}{\partial x} dx = \sigma(E...
...\delta E)\simeq
-\frac{\partial \sigma}{\partial E} \delta E,
\end{displaymath}">|; 

$key = q/{displaymath}frac{dH}{dt}=sum_rsum_sW_{sr},(P_r-P_s),(lnP_s+1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="\begin{displaymath}
\frac{d H}{dt} = \sum_r\sum_s W_{sr}  (P_r-P_s) (\ln P_s +1).
\end{displaymath}">|; 

$key = q/{displaymath}R_ast=frac{2,A}{B},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1614.png"
 ALT="\begin{displaymath}
R_\ast = \frac{2 A}{B},
\end{displaymath}">|; 

$key = q/V=V_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img603.png"
 ALT="$V=V_f$">|; 

$key = q/T_odotsimeq5770^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1511.png"
 ALT="$T_\odot\simeq 5770^\circ$">|; 

$key = q/AB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1370.png"
 ALT="$AB$">|; 

$key = q/asim(VslashN)^{1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1244.png"
 ALT="$a\sim (V/N)^{1/3}$">|; 

$key = q/{displaymath}E_0=frac{f,hbar^2pi^2}{2,m,L^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img695.png"
 ALT="\begin{displaymath}
E_0 = \frac{ f \hbar^2 \pi^2}{2 m  L^2}.
\end{displaymath}">|; 

$key = q/{displaymath}eta=frac{w}{q_1}=frac{T_1-T_2}{T_1}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img905.png"
 ALT="\begin{displaymath}
\eta = \frac{w}{q_1} = \frac{T_1 - T_2}{T_1}
\end{displaymath}">|; 

$key = q/displaystyleT',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img679.png"
 ALT="$\displaystyle T',$">|; 

$key = q/T=298^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$T= 298^\circ $">|; 

$key = q/bar{p},,d{{bf{S}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="$\bar{p}  d{\bf S}_i$">|; 

$key = q/{displaymath}[p(z)-p(z+dz)]A=rho,A,dz,g,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img807.png"
 ALT="\begin{displaymath}[p(z)- p(z+dz)]A = \rho  A dz g,
\end{displaymath}">|; 

$key = q/{displaymath}overline{M}_0simeqchi,B,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img965.png"
 ALT="\begin{displaymath}
\overline{M}_0 \simeq \chi  B,
\end{displaymath}">|; 

$key = q/{displaymath}overline{E}=-frac{partial}{partialbeta}lnZ=-left[-frac{1}{2},hbar,oega),hbar,omega}{1-exp(-beta,hbar,omega)}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1097.png"
 ALT="\begin{displaymath}
\overline{E} = - \frac{\partial}{\partial \beta} \ln Z = -
\...
...omega) \hbar \omega}
{1-\exp(-\beta \hbar \omega)}\right],
\end{displaymath}">|; 

$key = q/{displaymath}sigma_0=frac{3}{2},ln!left(frac{2,pi,m,k}{h_0^{~2}}right)+frac{5}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1061.png"
 ALT="\begin{displaymath}
\sigma_0 = \frac{3}{2} \ln\!\left(\frac{2 \pi  m  k}{h_0^{&nbsp;2}}\right) + \frac{5}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{as~}Trightarrow0_{+},~~~SrightarrowS_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img707.png"
 ALT="\begin{displaymath}
{\rm as&nbsp;} T\rightarrow 0_{+},&nbsp;&nbsp;&nbsp;S\rightarrow S_0,
\end{displaymath}">|; 

$key = q/{displaymath}n!=n(n-1)(n-2)cdots3cdot2cdot1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="\begin{displaymath}
n! = n(n-1)(n-2) \cdots 3\cdot 2 \cdot 1.
\end{displaymath}">|; 

$key = q/displaystyle0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1379.png"
 ALT="$\displaystyle 0.$">|; 

$key = q/1slash26;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="$1/26$">|; 

$key = q/chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img733.png"
 ALT="$\chi$">|; 

$key = q/{displaymath}sum_{n_1=0}^Nfrac{N!}{n_1!,(N-n_1)!},p^{n_1},q^{N-n_1},n_1equivp,fr{partial}{partialp},(p+q)^Nequivp,N,(p+q)^{N-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="528" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="\begin{displaymath}
\sum_{n_1=0}^N\frac{N!}{n_1! (N-n_1)!} p^{n_1} q^{N-n_1}\...
...\frac{\partial}{\partial p}  (p+q)^N\equiv p N (p+q)^{N-1}.
\end{displaymath}">|; 

$key = q/{mitOmega}^{(0)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img567.png"
 ALT="${\mit\Omega}^{(0)}$">|; 

$key = q/epsilon<mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1543.png"
 ALT="$\epsilon&lt;\mu$">|; 

$key = q/{mitOmega}^ast(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img631.png"
 ALT="${\mit\Omega}^\ast (E)$">|; 

$key = q/sigma(omega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1232.png"
 ALT="$\sigma(\omega)$">|; 

$key = q/{displaymath}c=frac{1}{nu}frac{{mitDelta}Q}{{mitDelta}T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img753.png"
 ALT="\begin{displaymath}
c = \frac{1}{\nu}\frac{{\mit\Delta} Q}{{\mit\Delta} T}.
\end{displaymath}">|; 

$key = q/mu_B=mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img943.png"
 ALT="$\mu_B = \mu$">|; 

$key = q/{displaymath}P_{{rm{tot}(T)=int_0^inftyP(omega),domega=frac{hbar}{4pi^2,c^2}int_frac{omega^3,domega}{exp(hbar,omegaslashk,T)-1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1503.png"
 ALT="\begin{displaymath}
P_{\rm tot}(T) = \int_0^\infty P(\omega)   d\omega
= \frac{...
...infty \frac{\omega^3 d\omega}
{\exp(\hbar \omega/k T) - 1},
\end{displaymath}">|; 

$key = q/lnP_r>lnP_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$\ln P_r &gt; \ln P_s$">|; 

$key = q/{displaymath}int_i^f{mathchar'26mskip-12mud}G=int_i^f(X',dx+Y',dy){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="\begin{displaymath}
\int_i^f {\mathchar'26\mskip-12mud}G = \int_i^f (X'  dx + Y'  dy)
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}_fgeq{mitOmega}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img597.png"
 ALT="\begin{displaymath}
{\mit\Omega}_f \geq {\mit\Omega}_i.
\end{displaymath}">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=dE.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img758.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = dE.
\end{displaymath}">|; 

$key = q/textstyleleftrightarrow;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1349.png"
 ALT="$\textstyle \leftrightarrow$">|; 

$key = q/{displaymath}E=E(T).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img734.png"
 ALT="\begin{displaymath}
E = E(T).
\end{displaymath}">|; 

$key = q/p+dp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1600.png"
 ALT="$p+dp$">|; 

$key = q/1slash6times1slash6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$1/6 \times 1/6$">|; 

$key = q/h_0^{~3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="$h_0^{&nbsp;3}$">|; 

$key = q/10^{10};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="$10^{10}$">|; 

$key = q/y_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="$y_i$">|; 

$key = q/displaystyleP(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$\displaystyle P(1)$">|; 

$key = q/{displaymath}overline{E}=3,N,hbar,omegaleft(frac{1}{2}+frac{1}{exp(beta,hbar,omega)-1}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1187.png"
 ALT="\begin{displaymath}
\overline{E} = 3 N  \hbar \omega\left(\frac{1}{2} +
\frac{1}{\exp(\beta \hbar \omega) - 1} \right).
\end{displaymath}">|; 

$key = q/{displaymath}dlnZ=beta,,{mathchar'26mskip-12mud}W-overline{E},dbeta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1002.png"
 ALT="\begin{displaymath}
d\ln Z = \beta   {\mathchar'26\mskip-12mud}W - \overline{E}  d\beta.
\end{displaymath}">|; 

$key = q/{displaymath}overline{u}(omega),domega=hbar,omega,,n(omega),domega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1490.png"
 ALT="\begin{displaymath}
\overline{u}(\omega) d\omega = \hbar \omega   n(\omega)  d\omega,
\end{displaymath}">|; 

$key = q/N-n_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$N-n_1$">|; 

$key = q/N!;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1054.png"
 ALT="$N!$">|; 

$key = q/rs;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1008.png"
 ALT="$rs$">|; 

$key = q/displaystyletimesint_{-infty}^{infty}exp[-(betaslash2m),p_z^{~2}],dp_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1028.png"
 ALT="$\displaystyle \times \int_{-\infty}^{\infty}\exp[-(\beta/2m) p_z^{&nbsp;2}] dp_z$">|; 

$key = q/{displaymath}S^{(0)}=kln{mitOmega}^{(0)}(tilde{E})=kln[{mitOmega}(tilde{E}),{mitln{mitOmega}(tilde{E})+kln{mitOmega}'(tilde{E}'){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="522" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img650.png"
 ALT="\begin{displaymath}
S^{(0)} = k\ln{\mit\Omega}^{(0)}(\tilde{E}) = k\ln[{\mit\Ome...
...=k\ln {\mit\Omega}(\tilde{E}) + k\ln {\mit\Omega}'(\tilde{E}')
\end{displaymath}">|; 

$key = q/6times5times4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$6\times 5\times 4$">|; 

$key = q/E^{(0)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="$E^{(0)}$">|; 

$key = q/{displaymath}dS=frac{1}{T},dE+frac{nu,R}{V},dV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img737.png"
 ALT="\begin{displaymath}
dS = \frac{1}{T}  dE+ \frac{\nu  R}{V}  dV.
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{k,T}=frac{partialln{mitOmega}}{partialE}=frac{3N}{2}frac{1}{E},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img779.png"
 ALT="\begin{displaymath}
\frac{1}{k T} = \frac{\partial \ln{\mit\Omega}}{\partial E} = \frac{3N}{2} \frac{1}{E},
\end{displaymath}">|; 

$key = q/hbar,omegallk,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1483.png"
 ALT="$\hbar \omega\ll k T$">|; 

$key = q/{displaymath}lnZ=alpha,Npmsum_rln,(1pm{{rm{e}^{-alpha-beta,epsilon_r}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1444.png"
 ALT="\begin{displaymath}
\ln Z = \alpha N \pm \sum_r \ln (1\pm {\rm e}^{-\alpha-\beta \epsilon_r}).
\end{displaymath}">|; 

$key = q/2,nu,R,ln2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1050.png"
 ALT="$2 \nu  R  \ln 2 $">|; 

$key = q/x=V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img996.png"
 ALT="$x=V$">|; 

$key = q/{displaymath}K=frac{3,V}{{mitLambda}^3}int_0^{p_F}frac{p^2}{2,m},p^2,dp=frac{3}{5},frac{V}{{mitLambda}^3},frac{p_F^{,5}}{2,m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1602.png"
 ALT="\begin{displaymath}
K = \frac{3 V}{{\mit\Lambda}^3}\int_0^{p_F} \frac{p^2}{2 m...
...frac{3}{5} \frac{V}{{\mit\Lambda}^3} \frac{p_F^{ 5}}{2 m},
\end{displaymath}">|; 

$key = q/{displaymath}Z=zeta^N,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1024.png"
 ALT="\begin{displaymath}
Z = \zeta^N,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cobe.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="521" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img1502.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cobe.eps}}
\end{figure}">|; 

$key = q/displaystylesum_{r,s}exp(-beta,[E_r+E_s']);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1012.png"
 ALT="$\displaystyle \sum_{r,s} \exp(-\beta [E_r+E_s'])$">|; 

$key = q/{displaymath}overline{M}_0=N_0,overline{mu}_B.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img964.png"
 ALT="\begin{displaymath}
\overline{M}_0 = N_0  \overline{\mu}_B.
\end{displaymath}">|; 

$key = q/{displaymath}C_Vsimeqfrac{3}{2},N_{{rm{eff},k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1578.png"
 ALT="\begin{displaymath}
C_V\simeq \frac{3}{2} N_{\rm eff} k.
\end{displaymath}">|; 

$key = q/p,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img890.png"
 ALT="$p  dV$">|; 

$key = q/{displaymath}rho_k,dk=frac{4pik^2,dk}{(2pi)^3}=frac{k^2}{2pi^2},dk.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1226.png"
 ALT="\begin{displaymath}
\rho_k  dk = \frac{4\pi k^2 dk}{(2\pi)^3} = \frac{k^2}{2 \pi^2}  dk.
\end{displaymath}">|; 

$key = q/{displaymath}C^N=N!,.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="\begin{displaymath}
C^N = N!  .
\end{displaymath}">|; 

$key = q/{mitDelta}E=hbar,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1181.png"
 ALT="${\mit\Delta} E = \hbar  \omega$">|; 

$key = q/k,Tll{mitDelta}E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1124.png"
 ALT="$k T \ll {\mit\Delta} E$">|; 

$key = q/int_i^f{mathchar'26mskip-12mud}QslashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img590.png"
 ALT="$\int_i^f {\mathchar'26\mskip-12mud}Q/T$">|; 

$key = q/<10^9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1126.png"
 ALT="$&lt;10^9$">|; 

$key = q/{displaymath}overline{({mitDelta}u)^2}=sum_{i=1}^MP(u_i),(u_i-bar{u})^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="\begin{displaymath}
\overline{({\mit\Delta} u)^2} = \sum_{i=1}^M P(u_i) (u_i - \bar{u})^2,
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{2},k,T.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1088.png"
 ALT="$\displaystyle \frac{1}{2}  k T.$">|; 

$key = q/T^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$T^2$">|; 

$key = q/{displaymath}tilde{v}=sqrt{frac{2,k,T}{m}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1335.png"
 ALT="\begin{displaymath}
\tilde{v} = \sqrt{\frac{2  k T}{m}}.
\end{displaymath}">|; 

$key = q/T=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1546.png"
 ALT="$T=0$">|; 

$key = q/p,V=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img866.png"
 ALT="$p V =$">|; 

$key = q/>5times10^9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1142.png"
 ALT="$&gt;5\times 10^9 $">|; 

$key = q/y=y_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img607.png"
 ALT="$y=y_i$">|; 

$key = q/{displaymath}eta=frac{373-288}{373}simeq0.23.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img909.png"
 ALT="\begin{displaymath}
\eta = \frac{373-288}{373} \simeq 0.23.
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{5},G,M^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1612.png"
 ALT="$\displaystyle \frac{3}{5} G M^2.$">|; 

$key = q/N_A,k=R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img781.png"
 ALT="$N_A  k=R$">|; 

$key = q/{displaymath}sum_rexp(-beta,E_r),frac{partialE_r}{partialx}=-frac{1}{beta}frac{pr)right]=-frac{1}{beta}frac{partialZ}{partialx},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="466" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img991.png"
 ALT="\begin{displaymath}
\sum_r \exp(-\beta  E_r) \frac{\partial E_r}{\partial x} =...
...,E_r)\right] = -\frac{1}{\beta}
\frac{\partial Z}{\partial x},
\end{displaymath}">|; 

$key = q/lambda_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1562.png"
 ALT="$\lambda_F$">|; 

$key = q/{displaymath}dS=frac{{mathchar'26mskip-12mud}Q}{T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1005.png"
 ALT="\begin{displaymath}
dS = \frac{{\mathchar'26\mskip-12mud}Q}{T}.
\end{displaymath}">|; 

$key = q/{displaymath}{{rm{Average~Age}simeqfrac{N_{18}times18+N_{19}times19+N_{20}times20+cdots}{N_{18}+N_{19}+N_{20}cdots},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="\begin{displaymath}
{\rm Average&nbsp;Age} \simeq \frac{N_{18}\times 18 + N_{19}\times 19 +N_{20}
\times 20+\cdots}
{N_{18}+N_{19}+N_{20}\cdots},
\end{displaymath}">|; 

$key = q/displaystyleeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="$\displaystyle \eta$">|; 

$key = q/displaystylehbar,{{bf{k},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1479.png"
 ALT="$\displaystyle \hbar {\bf k},$">|; 

$key = q/{displaymath}A^2=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1360.png"
 ALT="\begin{displaymath}
A^2 = 1.
\end{displaymath}">|; 

$key = q/{displaymath}p^{,1-gamma},T^{,gamma}={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img800.png"
 ALT="\begin{displaymath}
p^{ 1-\gamma}  T^{ \gamma} = {\rm constant}.
\end{displaymath}">|; 

$key = q/q_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1064.png"
 ALT="$q_k$">|; 

$key = q/0.85;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img837.png"
 ALT="$0.85$">|; 

$key = q/displaystyleleft(frac{partial^2S}{partialE^2}right)_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img682.png"
 ALT="$\displaystyle \left(\frac{\partial^2 S}{\partial E^2}\right)_V$">|; 

$key = q/{displaymath}~_{{{rm{lt},mrightarrow0}left(1+m,xright)^{1slashm}equivexp(x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img856.png"
 ALT="\begin{displaymath}
&nbsp;_{ {\rm lt}  m\rightarrow 0} \left(1 + m x\right)^{1/m} \equiv \exp(x).
\end{displaymath}">|; 

$key = q/s=1,2,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1369.png"
 ALT="$s=1,2,3$">|; 

$key = q/k,Tgghbar,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1107.png"
 ALT="$k T \gg \hbar  \omega$">|; 

$key = q/displaystyleP(2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$\displaystyle P(2)$">|; 

$key = q/h_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="$h_0$">|; 

$key = q/g(v_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1308.png"
 ALT="$g(v_z)$">|; 

$key = q/{displaymath}lnP(E)=lnP(tilde{E})-frac{1}{2},lambda_0,eta^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="\begin{displaymath}
\ln P(E) = \ln P(\tilde{E}) - \frac{1}{2} \lambda_0  \eta^2,
\end{displaymath}">|; 

$key = q/{{bf{v}+d{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1280.png"
 ALT="${\bf v}+d{\bf v}$">|; 

$key = q/{displaymath}P(E)=C,{mitOmega}(E),exp(-betaE),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img935.png"
 ALT="\begin{displaymath}
P(E) = C  {\mit\Omega}(E) \exp(-\beta E),
\end{displaymath}">|; 

$key = q/displaystylesum_{r,s}exp[-beta,E_{rs}^{(0)}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1011.png"
 ALT="$\displaystyle \sum_{r,s} \exp[-\beta  E_{rs}^{(0)}]$">|; 

$key = q/{displaymath}frac{dH}{dt}=sum_rsum_sW_{rs},(P_s-P_r),(lnP_r+1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="\begin{displaymath}
\frac{d H}{dt} = \sum_r\sum_s W_{rs}  (P_s-P_r) (\ln P_r +1).
\end{displaymath}">|; 

$key = q/bar{X}_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img559.png"
 ALT="$\bar{X}_\alpha$">|; 

$key = q/TggT_{{rm{rad};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1121.png"
 ALT="$T\gg T_{\rm rad}$">|; 

$key = q/{mitOmega}proptoE^f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="${\mit\Omega}\propto E^f$">|; 

$key = q/{displaymath}C^{-1}=sum_rexp(-beta,E_r),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img932.png"
 ALT="\begin{displaymath}
C^{-1} = \sum_r \exp(-\beta  E_r),
\end{displaymath}">|; 

$key = q/<0.05;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1127.png"
 ALT="$&lt;0.05$">|; 

$key = q/tau(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="$\tau(x, y)$">|; 

$key = q/n=tilde{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$n=\tilde{n}$">|; 

$key = q/{displaymath}c_V=frac{1}{nu}left(frac{{mathchar'26mskip-12mud}Q}{dT}right)_V,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img755.png"
 ALT="\begin{displaymath}
c_V = \frac{1}{\nu}\left(\frac{{\mathchar'26\mskip-12mud}Q}{dT}\right)_V,
\end{displaymath}">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}G=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}G = 0,
\end{displaymath}">|; 

$key = q/n_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1215.png"
 ALT="$n_y$">|; 

$key = q/0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img829.png"
 ALT="$0.5$">|; 

$key = q/y_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="$y_k$">|; 

$key = q/epsilon^{{rm{int};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1269.png"
 ALT="$\epsilon^{\rm int}$">|; 

$key = q/10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img701.png"
 ALT="$10^{-3}$">|; 

$key = q/displaystyleleft(frac{partialS}{partialV}right)_{E},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img673.png"
 ALT="$\displaystyle \left(\frac{\partial S}{\partial V}\right)_{E},$">|; 

$key = q/R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$R$">|; 

$key = q/{displaymath}q_1=int_a^bnu,R,T_1frac{dV}{V}=nu,R,T_1,lnfrac{V_b}{V_a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img896.png"
 ALT="\begin{displaymath}
q_1 = \int_a^b \nu R  T_1\frac{ dV}{V} = \nu  R  T_1  \ln\frac{V_b}{V_a}.
\end{displaymath}">|; 

$key = q/0.3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img836.png"
 ALT="$0.3$">|; 

$key = q/displaystylebar{E}_f'-E_i'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img515.png"
 ALT="$\displaystyle \bar{E}_f' - E_i'.$">|; 

$key = q/bar{E}(T);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1569.png"
 ALT="$\bar{E}(T)$">|; 

$key = q/{displaymath}E=frac{A}{R^2}-frac{B}{R},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1608.png"
 ALT="\begin{displaymath}
E = \frac{A}{R^2} - \frac{B}{R},
\end{displaymath}">|; 

$key = q/{displaymath}{calP}(n)simeq{calP}(overline{n_1})exp!left[-frac{(n-overline{n_1})^2}{2,({mitDelta}^astn_1)^2}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="\begin{displaymath}
{\cal P}(n)\simeq {\cal P}(\overline{n_1})\exp\!\left[-
\frac{(n-\overline{n_1})^2}{2 ({\mit\Delta}^\ast n_1)^2}\right].
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{5},G,M^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1627.png"
 ALT="$\displaystyle \frac{3}{5} G M^2,$">|; 

$key = q/{displaymath}overline{E}simeqfrac{1}{beta}=k,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1103.png"
 ALT="\begin{displaymath}
\overline{E} \simeq \frac{1}{\beta} = k T.
\end{displaymath}">|; 

$key = q/W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="$W$">|; 

$key = q/{mitDelta}bar{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="${\mit\Delta} \bar{E}$">|; 

$key = q/{displaymath}P_N(n_1)=C^{N}_{n_1,,N-n_1},p^{n_1},q^{N-n_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="\begin{displaymath}
P_N(n_1) = C^{ N}_{n_1, N-n_1} p^{n_1} 
q^{N-n_1},
\end{displaymath}">|; 

$key = q/{displaymath}C_V=left(frac{partialbar{E}}{partialT}right)_V.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1570.png"
 ALT="\begin{displaymath}
C_V = \left(\frac{\partial \bar{E}}{\partial T}\right)_V.
\end{displaymath}">|; 

$key = q/{displaymath}p^{,1-gamma},T^{,gamma}={{rm{constant},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img838.png"
 ALT="\begin{displaymath}
p^{ 1-\gamma}  T^{ \gamma} = {\rm constant},
\end{displaymath}">|; 

$key = q/tilde{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="$\tilde{E}$">|; 

$key = q/p_{i,y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="$p_{i y}$">|; 

$key = q/{displaymath}(c_V+R),p,dV+c_V,V,dp=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img792.png"
 ALT="\begin{displaymath}
(c_V +R) p dV + c_V  V  dp = 0.
\end{displaymath}">|; 

$key = q/(3slash2),N,k,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1062.png"
 ALT="$(3/2) N
 k T$">|; 

$key = q/{mitOmega}(E;y_k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="${\mit\Omega}(E; y_k)$">|; 

$key = q/{displaymath}{mitOmega}proptoV^f.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img568.png"
 ALT="\begin{displaymath}
{\mit\Omega} \propto V^f.
\end{displaymath}">|; 

$key = q/{displaymath}overline{epsilon}_i=-frac{partial}{partialbeta}lnleft[int_{-infty}^inftyexp(-beta,epsilon_i),dp_iright].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1077.png"
 ALT="\begin{displaymath}
\overline{\epsilon}_i = - \frac{\partial}{\partial\beta}
\l...
...int_{-\infty}^\infty \exp(-\beta  \epsilon_i) 
dp_i\right].
\end{displaymath}">|; 

$key = q/{displaymath}overline{frac{1}{2},m,v^2}=overline{frac{1}{2},m,(v_x^{~2}+v_y^{~2}+v_z^{~2})}=3left(frac{1}{2},k,Tright),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1334.png"
 ALT="\begin{displaymath}
\overline{\frac{1}{2} m v^2} = \overline{\frac{1}{2} m (v_x^{&nbsp;2}+v_y^{&nbsp;2}+v_z^{&nbsp;2})}
= 3 \left(\frac{1}{2}  k T\right),
\end{displaymath}">|; 

$key = q/Z_s(N);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1398.png"
 ALT="$Z_s(N)$">|; 

$key = q/{displaymath}{mitOmega}(XmidY)={mitOmega}(X)+{mitOmega}(Y){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="\begin{displaymath}
{\mit\Omega}(X\mid Y) = {\mit\Omega}(X) + {\mit\Omega}(Y)
\end{displaymath}">|; 

$key = q/{displaymath}sigma=frac{pi^2}{60}frac{k^4}{c^2,hbar^3}=5.67times10^{-8},,{{rm{W},{{rm{m}^{-2},{{rm{K}^{-4},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1509.png"
 ALT="\begin{displaymath}
\sigma = \frac{\pi^2}{60} \frac{k^4}{c^2  \hbar^3} = 5.67\times 10^{-8}  {\rm W}
 {\rm m}^{-2}  {\rm K}^{-4},
\end{displaymath}">|; 

$key = q/{displaymath}v_{{rm{rms}=sqrt{overline{v^2}}=sqrt{frac{3,k,T}{m}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1333.png"
 ALT="\begin{displaymath}
v_{\rm rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3  k T}{m}}.
\end{displaymath}">|; 

$key = q/{displaymath}overline{E^2}=frac{partial}{partialbeta}!left(frac{1}{Z}frac{partia{partialoverline{E}}{partialbeta}+overline{E}^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img985.png"
 ALT="\begin{displaymath}
\overline{ E^2} = \frac{\partial}{\partial \beta}\!\left(
\f...
...frac{\partial \overline{E}}{\partial
\beta} + \overline{E}^2,
\end{displaymath}">|; 

$key = q/{displaymath}eta=frac{277}{288-277}=25.2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img917.png"
 ALT="\begin{displaymath}
\eta = \frac{277}{288-277} = 25.2.
\end{displaymath}">|; 

$key = q/n_{i2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="$n_{i2}$">|; 

$key = q/{displaymath}f({{bf{v}),d^3{{bf{r},d^3{{bf{v}=nleft(frac{m}{2pi,k,T}right)^{3slah2},exp(-m,v^2slash2,k,T),d^3{{bf{r},d^3{{bf{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1290.png"
 ALT="\begin{displaymath}
f({\bf v})  d^3{\bf r} d^3{\bf v} = n \left(\frac{m}{2\pi ...
...ht)^{3/2}
  \exp(-m  v^2/ 2 k T)  d^3{\bf r} d^3{\bf v}.
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{2},N,k,T,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1573.png"
 ALT="$\displaystyle \frac{3}{2} N k T,$">|; 

$key = q/E'=E^{(0)}-E_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img920.png"
 ALT="$E'=E^{(0)} - E_r$">|; 

$key = q/A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="$A$">|; 

$key = q/{mathchar'26mskip-12mud}W=p,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img724.png"
 ALT="${\mathchar'26\mskip-12mud}W =p dV$">|; 

$key = q/displaystylebeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="$\displaystyle \beta$">|; 

$key = q/{displaymath}E=E(q_1,cdots,q_f,p_1,cdots,p_f).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1066.png"
 ALT="\begin{displaymath}
E = E(q_1, \cdots, q_f, p_1,\cdots, p_f).
\end{displaymath}">|; 

$key = q/displaystyle{{rm{for~}omega<omega_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1237.png"
 ALT="$\displaystyle {\rm for&nbsp;} \omega&lt;\omega_D$">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$i$">|; 

$key = q/{displaymath}gammaequivfrac{c_p}{c_V}=frac{c_V+R}{c_V}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img795.png"
 ALT="\begin{displaymath}
\gamma \equiv \frac{c_p}{c_V} = \frac{c_V + R}{c_V}.
\end{displaymath}">|; 

$key = q/h_0^{~f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="$h_0^{&nbsp;f}$">|; 

$key = q/{displaymath}E_R=sum_r,n_r,epsilon_r,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1382.png"
 ALT="\begin{displaymath}
E_R = \sum_r n_r \epsilon_r,
\end{displaymath}">|; 

$key = q/{displaymath}S^{(0)}=kln{mitOmega}^{(0)}_{{rm{tot}=kln{mitOmega}^{(0)}(tilde{E})+klnfrac{{mitDelta}^astE}{deltaE}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img647.png"
 ALT="\begin{displaymath}
S^{(0)} = k \ln {\mit\Omega}^{(0)}_{\rm tot} = k\ln{\mit\Omega}^{(0)}(\tilde{E})+
k\ln \frac{{\mit\Delta}^\ast E}{\delta E}.
\end{displaymath}">|; 

$key = q/n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$n$">|; 

$key = q/{mitOmega}_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img596.png"
 ALT="${\mit\Omega}_f$">|; 

$key = q/displaystyleQ_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1348.png"
 ALT="$\displaystyle Q_i$">|; 

$key = q/{displaymath}xiequivfrac{mbox{probabilitythatthetwoparticlesarefoundinthesamestaythatthetwoparticlesarefoundindifferentstates}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="553" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1373.png"
 ALT="\begin{displaymath}
\xi \equiv \frac{\mbox{ probability that the two particles a...
...bility that the two particles are found in different
states}}.
\end{displaymath}">|; 

$key = q/{displaymath}P_{-}=Cexp(-beta,epsilon_{-})=Cexp(-beta,mu,B).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img948.png"
 ALT="\begin{displaymath}
P_{-} = C\exp(-\beta \epsilon_{-}) = C\exp(-\beta  \mu  B).
\end{displaymath}">|; 

$key = q/{displaymath}(c_V)_{{rm{rotation}=frac{3}{2},R,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1157.png"
 ALT="\begin{displaymath}
(c_V)_{\rm rotation} = \frac{3}{2}  R,
\end{displaymath}">|; 

$key = q/displaystylek_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1209.png"
 ALT="$\displaystyle k_x$">|; 

$key = q/tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="$\tau$">|; 

$key = q/2,f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="$2 f$">|; 

$key = q/{displaymath}frac{partialln{mitOmega}}{partialx}=frac{partialln{mitOmega}}{partialE},bar{X}=beta,bar{X}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img554.png"
 ALT="\begin{displaymath}
\frac{\partial\ln {\mit\Omega}}{\partial x} = \frac{\partial \ln {\mit\Omega}}{\partial E}
 \bar{X} = \beta  \bar{X}.
\end{displaymath}">|; 

$key = q/P_r>P_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="$P_r &gt; P_s$">|; 

$key = q/T_{{rm{atmosphere}>T_{{rm{packet};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img848.png"
 ALT="$T_{\rm atmosphere} &gt; T_{\rm packet}$">|; 

$key = q/B_1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$B_1=0$">|; 

$key = q/{displaymath}sum_{n=0}^inftyexp(-n,beta,hbar,omega)=1+exp(-beta,hbar,omega)+exp(-2,beta,hbar,omega)+cdots{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="477" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1093.png"
 ALT="\begin{displaymath}
\sum_{n=0}^\infty \exp(- n \beta  \hbar  \omega) = 1 + \e...
...beta \hbar \omega)
+ \exp(-2 \beta \hbar \omega) + \cdots
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}^astE=lambda_0^{-1slash2}simfrac{bar{E}}{sqrt{f}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="\begin{displaymath}
{\mit\Delta}^\ast E = \lambda_0^{-1/2}\sim \frac{\bar{E}}{\sqrt{f}},
\end{displaymath}">|; 

$key = q/{{rm{He}^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1366.png"
 ALT="${\rm He}^4$">|; 

$key = q/asim10^{-5},d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1160.png"
 ALT="$a \sim 10^{-5}  d$">|; 

$key = q/{displaymath}frac{1}{sqrt{N}}ll1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="\begin{displaymath}
\frac{1}{\sqrt{N}} \ll 1,
\end{displaymath}">|; 

$key = q/q_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="$q_y$">|; 

$key = q/displaystyleleft(p,frac{partial}{partialp}right)left[p,N,(p+q)^{N-1}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\displaystyle \left(p \frac{\partial}{\partial p}\right)\left[p N  (p+q)^{N-1}\right]$">|; 

$key = q/{displaymath}K=frac{3}{5},N,frac{{mitLambda}^2}{2,m}left(frac{N}{V}right)^{2slash3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1603.png"
 ALT="\begin{displaymath}
K = \frac{3}{5} N \frac{{\mit\Lambda}^2}{2 m}\left(\frac{N}{V}\right)^{2/3}.
\end{displaymath}">|; 

$key = q/N_{{rm{students};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$N_{\rm students}$">|; 

$key = q/{displaymath}V+V'=V^{(0)}={{rm{constant},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img564.png"
 ALT="\begin{displaymath}
V + V' = V^{(0)} = {\rm constant},
\end{displaymath}">|; 

$key = q/sum_rn_r=N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1426.png"
 ALT="$\sum_r n_r=N$">|; 

$key = q/2pi,c_sslashomega_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1243.png"
 ALT="$2\pi c_s/\omega_D$">|; 

$key = q/{displaymath}T_oplus=left(frac{R_odot}{2,r_oplus}right)^{1slash2}T_odot.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1521.png"
 ALT="\begin{displaymath}
T_\oplus = \left(\frac{R_\odot}{2 r_\oplus}\right)^{1/2} T_\odot.
\end{displaymath}">|; 

$key = q/bar{u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\bar{u}$">|; 

$key = q/43^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1524.png"
 ALT="$43^\circ$">|; 

$key = q/P(XotimesY);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$P(X\otimes Y)$">|; 

$key = q/4,p,q^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$4 p q^3$">|; 

$key = q/displaystyleZ^{(0)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1010.png"
 ALT="$\displaystyle Z^{(0)}$">|; 

$key = q/y_f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="$y_f)$">|; 

$key = q/T^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1264.png"
 ALT="$T^3$">|; 

$key = q/displaystylebar{p}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img681.png"
 ALT="$\displaystyle \bar{p}',$">|; 

$key = q/z+dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img801.png"
 ALT="$z+dz$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W=p,dV,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img716.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W = p dV,
\end{displaymath}">|; 

$key = q/lambda(tilde{E});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="$\lambda(\tilde{E})$">|; 

$key = q/exp[-{{rm{i},({{bf{k}!cdot{{bf{r}-omega,t)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1198.png"
 ALT="$\exp[-{\rm i} ({\bf k}\!\cdot{\bf r} - \omega  t)]$">|; 

$key = q/{displaymath}{mitOmega}proptoV^Nchi(E),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img718.png"
 ALT="\begin{displaymath}
{\mit\Omega} \propto V^N \chi(E),
\end{displaymath}">|; 

$key = q/lambda=2pislashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1467.png"
 ALT="$\lambda = 2\pi /k$">|; 

$key = q/{displaymath}z_0=frac{R,T_0}{mu,g}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img851.png"
 ALT="\begin{displaymath}
z_0 = \frac{R  T_0}{\mu  g}
\end{displaymath}">|; 

$key = q/P(Y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$P(Y)$">|; 

$key = q/{displaymath}eta=frac{T_1-T_2}{T_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img887.png"
 ALT="\begin{displaymath}
\eta = \frac{T_1 - T_2}{T_1},
\end{displaymath}">|; 

$key = q/1slash2,3slash2,5slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1365.png"
 ALT="$1/2, 3/2, 5/2$">|; 

$key = q/C^N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$C^N$">|; 

$key = q/{displaymath}Z=frac{Z_{{rm{MB}}{N!},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1459.png"
 ALT="\begin{displaymath}
Z = \frac{Z_{\rm MB}}{N!},
\end{displaymath}">|; 

$key = q/{mitOmega}proptoE^fV^f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img684.png"
 ALT="${\mit\Omega}\propto
E^f V^f$">|; 

$key = q/f+g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="$f+g$">|; 

$key = q/dsigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$d\sigma$">|; 

$key = q/1slash52;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="$1/52$">|; 

$key = q/{displaymath}c_V=frac{3,V,k}{2pi^2,nu,(c_s,beta,hbar)^3}int_0^{beta,hbar,omega_D}frac{expx}{(expx-1)^2},x^4,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1249.png"
 ALT="\begin{displaymath}
c_V = \frac{3 V  k}{2\pi^2  \nu (c_s  \beta  \hbar)^3}...
...eta \hbar \omega_D}
\frac{\exp x}{(\exp x - 1)^2} x^4 dx,
\end{displaymath}">|; 

$key = q/{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="${\bf p}$">|; 

$key = q/T_oplussim279^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1522.png"
 ALT="$T_\oplus\sim 279^\circ$">|; 

$key = q/{{mitSigma}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="${{\mit\Sigma}}$">|; 

$key = q/{displaymath}p=p_0left(1-frac{z}{3.5,z_0}right)^{3.5}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img858.png"
 ALT="\begin{displaymath}
p= p_0 \left(1 - \frac{z}{3.5 z_0} \right)^{3.5}.
\end{displaymath}">|; 

$key = q/displaystylesum_{r,s}exp(-beta,E_r)exp(-beta,E_s');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1013.png"
 ALT="$\displaystyle \sum_{r,s} \exp(-\beta  E_r) \exp(-\beta  E_s')$">|; 

$key = q/{displaymath}overline{M}_0rightarrowN_0,mu,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img968.png"
 ALT="\begin{displaymath}
\overline{M}_0 \rightarrow N_0  \mu,
\end{displaymath}">|; 

$key = q/x=beta,hbar,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1250.png"
 ALT="$x=\beta \hbar \omega$">|; 

$key = q/288^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img908.png"
 ALT="$288^\circ$">|; 

$key = q/E';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="$E'$">|; 

$key = q/{displaymath}muequiv-k,T,,alpha{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1526.png"
 ALT="\begin{displaymath}
\mu \equiv -k T  \alpha
\end{displaymath}">|; 

$key = q/V=V_f=2,V_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img601.png"
 ALT="$V=V_f=2  V_i$">|; 

$key = q/{displaymath}Z=sum_R{{rm{e}^{-beta,E_R}=sum_R{{rm{e}^{-beta,(n_1,epsilon_1+n_2,epsilon_2+cdots)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1384.png"
 ALT="\begin{displaymath}
Z =\sum_R {\rm e}^{-\beta E_R} = \sum_R {\rm e}^{-\beta (n_1 \epsilon_1+n_2 
\epsilon_2+\cdots)}.
\end{displaymath}">|; 

$key = q/V';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img565.png"
 ALT="$V'$">|; 

$key = q/S=S(T,V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img739.png"
 ALT="$S=S(T,V)$">|; 

$key = q/{displaymath}{mitDelta}uequivu-bar{u}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="\begin{displaymath}
{\mit\Delta} u \equiv u- \bar{u}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{nu}left(frac{partialoverline{E}}{partialT}right)_V=frac{1}{rtial[(1slash2),nu,R,T]}{partialT}=frac{1}{2},R,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1114.png"
 ALT="\begin{displaymath}
\frac{1}{\nu}\left(
\frac{\partial \overline{E}}{\partial T}...
...{\partial[ (1/2)  \nu  R T] }{\partial T}= \frac{1}{2}  R,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{setbox{0=hbox{{epsffile{maxw.eps}{}{centerline{rotu0}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="349" BORDER="0"
 SRC="|."$dir".q|img1327.png"
 ALT="\begin{figure}
\epsfysize =3in\setbox 0=\hbox{\epsffile{maxw.eps}}
\centerline{\rotu0}
\end{figure}">|; 

$key = q/{displaymath}0=frac{c_V}{R}(p,dV+V,dp)+p,dV=left(frac{c_V}{R}+1right),p,dV+frac{c_V}{R},V,dp,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="469" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img791.png"
 ALT="\begin{displaymath}
0 = \frac{c_V}{R} (p dV + V dp) + p  dV = \left(\frac{c_V}{R} + 1\right)
 p  dV +\frac{c_V}{R}  V dp,
\end{displaymath}">|; 

$key = q/Tlltheta_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1194.png"
 ALT="$T\ll \theta_E$">|; 

$key = q/F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="$F$">|; 

$key = q/{displaymath}frac{dlnn!}{dn}simeqlnn,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="\begin{displaymath}
\frac{d\ln n!}{d n} \simeq \ln n,
\end{displaymath}">|; 

$key = q/q_2<q_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img880.png"
 ALT="$q_2 &lt; q_1$">|; 

$key = q/(4slash3),pi,k_F^{,3},(2pi)^{-3},V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1558.png"
 ALT="$(4/3) \pi k_F^{ 3} (2\pi)^{-3} V$">|; 

$key = q/{displaymath}{mitPsi}(cdotsQ_icdotsQ_jcdots)=+{mitPsi}(cdotsQ_jcdotsQ_icdots),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1362.png"
 ALT="\begin{displaymath}
{\mit\Psi}(\cdots Q_i\cdots Q_j\cdots)=
+{\mit\Psi}(\cdots Q_j\cdots Q_i\cdots),
\end{displaymath}">|; 

$key = q/10^{14};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1131.png"
 ALT="$10^{14}$">|; 

$key = q/{displaymath}rho=rho_0left(1-frac{gamma-1}{gamma}frac{z}{z_0}right)^{1slash(gamma-1)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img860.png"
 ALT="\begin{displaymath}
\rho= \rho_0 \left(1 - \frac{\gamma -1}{\gamma} \frac{z}{z_0} \right)^{1/(\gamma-1)},
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}Sgeq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img870.png"
 ALT="\begin{displaymath}
{\mit\Delta} S \geq 0.
\end{displaymath}">|; 

$key = q/{displaymath}theta=frac{theta_1,C_1+theta_2,C_2}{C_1+C_2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="\begin{displaymath}
\theta = \frac{\theta_1  C_1 + \theta_2  C_2}{C_1 + C_2},
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{infty}exp(-beta,epsilon_i),dp_i=int_{-infty}^{infty}e}{sqrt{beta}}int_{-infty}^{infty}exp(-b,y^2),dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="556" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1078.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} \exp(-\beta  \epsilon_i)  dp_i =
...
...1}{\sqrt{\beta}} \int_{-\infty}^{\infty} \exp(- b  y^2)  dy,
\end{displaymath}">|; 

$key = q/{displaymath}sum_{n_1=0}^NP_N(n_1)=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="\begin{displaymath}
\sum_{n_1=0}^N P_N(n_1)=1,
\end{displaymath}">|; 

$key = q/{displaymath}overline{p},V=N,k,T=nu,R,T,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1033.png"
 ALT="\begin{displaymath}
\overline{p}  V = N  k  T = \nu  R  T,
\end{displaymath}">|; 

$key = q/p,V=nu,R,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img811.png"
 ALT="$p V= \nu  R T$">|; 

$key = q/{displaymath}P,V=nu,R,T,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="\begin{displaymath}
P V =\nu  R T,
\end{displaymath}">|; 

$key = q/d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img897.png"
 ALT="$d$">|; 

$key = q/,{mathchar'26mskip-12mud};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="$ {\mathchar'26\mskip-12mud}$">|; 

$key = q/{displaymath}{calP}(n)simeqfrac{1}{sqrt{2pi},{mitDelta}^astn_1},exp!left[-frac{(verline{n_1})^2}{2,({mitDelta}^astn_1)^2}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="\begin{displaymath}
{\cal P}(n) \simeq \frac{1}{\sqrt{2\pi}  {\mit\Delta}^\ast ...
...ac{(n-\overline{n_1})^2}{2 ({\mit\Delta}^\ast n_1)^2}\right].
\end{displaymath}">|; 

$key = q/x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img562.png"
 ALT="$x'$">|; 

$key = q/{displaymath}k=1.381times10^{-23}~~{{rm{joulesslashkelvin},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img530.png"
 ALT="\begin{displaymath}
k = 1.381\times 10^{-23}&nbsp;&nbsp;{\rm joules / kelvin},
\end{displaymath}">|; 

$key = q/lambda'(tilde{E}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="$\lambda'(\tilde{E}')$">|; 

$key = q/{figure}{epsfysize{=3in{setbox{0=hbox{{epsffile{hcl.eps}{}{centerline{rotu0}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="548" HEIGHT="349" BORDER="0"
 SRC="|."$dir".q|img1167.png"
 ALT="\begin{figure}
\epsfysize =3in\setbox 0=\hbox{\epsffile{hcl.eps}}
\centerline{\rotu0}
\end{figure}">|; 

$key = q/displaystyleB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1611.png"
 ALT="$\displaystyle B$">|; 

$key = q/{displaymath}{mitOmega}^{(0)}_{{rm{tot}simeqfrac{{mitOmega}^{(0)}(tilde{E})}{deltaE},{mitDelta}^astE.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img646.png"
 ALT="\begin{displaymath}
{\mit\Omega}^{(0)}_{\rm tot} \simeq \frac{{\mit\Omega}^{(0)}(\tilde{E})}
{\delta E}  {\mit\Delta}^\ast E.
\end{displaymath}">|; 

$key = q/lambda_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="$\lambda_0$">|; 

$key = q/2,nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1045.png"
 ALT="$2 \nu$">|; 

$key = q/{displaymath}frac{partialX}{partialy}=frac{partialY}{partialx}=frac{partial^2F}{partialxpartialy}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="\begin{displaymath}
\frac{\partial X}{\partial y} = \frac{\partial Y}{\partial x}= \frac{\partial^2 F}
{\partial x \partial y}.
\end{displaymath}">|; 

$key = q/(x_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="$(x_i$">|; 

$key = q/expx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1257.png"
 ALT="$\exp x$">|; 

$key = q/p,V^{,gamma}=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img867.png"
 ALT="$p V^{ \gamma}=$">|; 

$key = q/displaystyledp_{i,x},dp_{i,y},dp_{i,z},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="$\displaystyle dp_{i x} dp_{i y}  dp_{i z},$">|; 

$key = q/S^ast=kln{mitOmega}^ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img634.png"
 ALT="$S^\ast = k\ln{\mit\Omega}^\ast$">|; 

$key = q/{displaymath}(N-tilde{n}),p=tilde{n},q,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="\begin{displaymath}
(N-\tilde{n})  p = \tilde{n} q,
\end{displaymath}">|; 

$key = q/bar{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1325.png"
 ALT="$\bar{c}$">|; 

$key = q/,{mathchar'26mskip-12mud}Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$ {\mathchar'26\mskip-12mud}Q$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="$f$">|; 

$key = q/AA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1372.png"
 ALT="$AA$">|; 

$key = q/delta^astE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img630.png"
 ALT="$\delta^\ast E$">|; 

$key = q/{displaymath}lnZ_s(N-{mitDelta}N)simeqlnZ_s(N)-frac{partiallnZ_s}{partialN},{mitDelta}N=lnZ_s(N)-alpha_s,{mitDelta}N,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="513" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1401.png"
 ALT="\begin{displaymath}
\ln Z_s(N-{\mit\Delta}N) \simeq \ln Z_s(N) - \frac{\partial ...
...al N} 
{\mit\Delta} N = \ln Z_s(N) - \alpha_s {\mit\Delta}N,
\end{displaymath}">|; 

$key = q/N_{18};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$N_{18}$">|; 

$key = q/F=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1542.png"
 ALT="$F=1$">|; 

$key = q/{displaymath}epsilon_i(p_i)=b,p_i^{~2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1069.png"
 ALT="\begin{displaymath}
\epsilon_i(p_i) = b p_i^{&nbsp;2},
\end{displaymath}">|; 

$key = q/k,Tllmu,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img961.png"
 ALT="$k  T \ll \mu  B$">|; 

$key = q/P(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img608.png"
 ALT="$P(y)$">|; 

$key = q/10^{-12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$10^{-12}$">|; 

$key = q/y_ineqtilde{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img613.png"
 ALT="$y_i\neq \tilde{y}$">|; 

$key = q/beta,epsilon_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1429.png"
 ALT="$\beta \epsilon_s$">|; 

$key = q/beta,(epsilon-mu)gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1532.png"
 ALT="$\beta (\epsilon-\mu)\gg 1$">|; 

$key = q/T_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1518.png"
 ALT="$T_\oplus$">|; 

$key = q/p(z),A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img802.png"
 ALT="$p(z) A$">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$c$">|; 

$key = q/g(v_x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1306.png"
 ALT="$g(v_x)$">|; 

$key = q/{displaymath}{mitDelta}S=frac{-q_1}{T_1}+frac{q_2}{T_2}geq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img881.png"
 ALT="\begin{displaymath}
{\mit\Delta} S = \frac{-q_1}{T_1} + \frac{q_2}{T_2} \geq 0.
\end{displaymath}">|; 

$key = q/eta=hbar,omegaslashk,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1505.png"
 ALT="$\eta = \hbar \omega/k  T$">|; 

$key = q/q^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$q^4$">|; 

$key = q/{displaymath}{mitOmega}=frac{1}{h_0^{~f}}intcdotsintdq_1cdotsdq_f,dp_1cdotsdp_f,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img686.png"
 ALT="\begin{displaymath}
{\mit\Omega} = \frac{1}{h_0^{&nbsp;f}} \int \cdots \int dq_1\cdots dq_f dp_1\cdots dp_f,
\end{displaymath}">|; 

$key = q/{displaymath}P(X)=~_{lt,{mitOmega}({mitSigma})rightarrowinfty}frac{{mitOmega}(X)}{{mitOmega}({mitSigma})},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="\begin{displaymath}
P(X) =&nbsp;_{lt {\mit\Omega}({\mit\Sigma})\rightarrow\infty}\frac{{\mit\Omega}(X)}{{\mit\Omega}({\mit\Sigma})},
\end{displaymath}">|; 

$key = q/{displaymath}int_0^inftyy^3exp(-y^2),dy=frac{1}{2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1331.png"
 ALT="\begin{displaymath}
\int_0^\infty y^3 \exp(-y^2) dy = \frac{1}{2},
\end{displaymath}">|; 

$key = q/{displaymath}Z_s(N-{mitDelta}N)=Z_s(N),{{rm{e}^{-alpha_s,{mitDelta}N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1404.png"
 ALT="\begin{displaymath}
Z_s(N-{\mit\Delta}N) = Z_s(N) {\rm e}^{-\alpha_s {\mit\Delta}N}.
\end{displaymath}">|; 

$key = q/{displaymath}sigma(E,x)=-frac{{mitOmega}(E,x)}{deltaE},bar{X},,dx,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img545.png"
 ALT="\begin{displaymath}
\sigma (E,x) = - \frac{{\mit\Omega}(E,x)}{\delta E} 
\bar{X}  dx,
\end{displaymath}">|; 

$key = q/8.314;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img821.png"
 ALT="$8.314$">|; 

$key = q/overline{mu}_Bsimeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img951.png"
 ALT="$\overline{\mu}_B \simeq 0$">|; 

$key = q/q_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img878.png"
 ALT="$q_2$">|; 

$key = q/3,p,q^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$3 p q^2$">|; 

$key = q/n_1pm1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$n_1\pm 1/2$">|; 

$key = q/k_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1217.png"
 ALT="$k_x$">|; 

$key = q/{displaymath}c_V=frac{k}{nu}int_0^{omega_D}frac{exp(beta,hbar,omega),(beta,hbar,-1]^2},frac{3,V}{2pi^2,c_s^{~3}},omega^2,domega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1248.png"
 ALT="\begin{displaymath}
c_V = \frac{k}{\nu}\int_0^{\omega_D} \frac{\exp(\beta \hbar...
...a)-1]^2} 
\frac{3  V}{2\pi^2  c_s^{&nbsp;3}} \omega^2 d\omega,
\end{displaymath}">|; 

$key = q/v+dv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1312.png"
 ALT="$v+dv$">|; 

$key = q/{displaymath}bar{n}_s=frac{0+{{rm{e}^{-beta,epsilon_s},Z_s(N-1)}{Z_s(N)+{{rm{e}^{-beta,epsilon_s},Z_s(N-1)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1395.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{0 + {\rm e}^{-\beta \epsilon_s} Z_s(N-1)}
{Z_s(N) + {\rm e}^{-\beta \epsilon_s} Z_s(N-1)},
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{T}left(frac{partialE}{partialT}right)_V,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img743.png"
 ALT="$\displaystyle \frac{1}{T} \left(\frac{\partial E}
{\partial T}\right)_V,$">|; 

$key = q/x_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img557.png"
 ALT="$x_n)$">|; 

$key = q/{displaymath}rho=frac{nu,mu}{V},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img809.png"
 ALT="\begin{displaymath}
\rho = \frac{\nu  \mu}{V},
\end{displaymath}">|; 

$key = q/k,Tgg{mitDelta}E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1122.png"
 ALT="$k T \gg {\mit\Delta} E$">|; 

$key = q/rho,Tproptop;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img846.png"
 ALT="$\rho  T \propto p$">|; 

$key = q/{displaymath}Ksimeqfrac{3,V,c}{{mitLambda}^3}int_0^{p_F}left(p^3+frac{m^2,c^2}{2},p+cdotsright)dp,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1623.png"
 ALT="\begin{displaymath}
K\simeq \frac{3 V c}{{\mit\Lambda}^3}\int_0^{p_F} \left(p^3
+ \frac{m^2 c^2}{2} p + \cdots\right)dp,
\end{displaymath}">|; 

$key = q/X_{alpha,r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$X_{\alpha r}$">|; 

$key = q/R_oplussim6000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1513.png"
 ALT="$R_\oplus\sim 6000$">|; 

$key = q/{displaymath}bar{n}_s=frac{sum_sn_s,{{rm{e}^{-n_s,(alpha+beta,epsilon_s)}}{sum_s{{rm{e}^{-n_s,(alpha+beta,epsilon_s)}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1428.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{\sum_s n_s {\rm e}^{-n_s (\alpha+\beta ...
...ilon_s)}}
{\sum_s {\rm e}^{-n_s (\alpha+\beta \epsilon_s)}}.
\end{displaymath}">|; 

$key = q/{displaymath}sum_rP_r=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img924.png"
 ALT="\begin{displaymath}
\sum_r P_r = 1,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{setbox{0=hbox{{epsffile{h2.eps}{}{centerline{rotu0}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="504" HEIGHT="349" BORDER="0"
 SRC="|."$dir".q|img1177.png"
 ALT="\begin{figure}
\epsfysize =3in\setbox 0=\hbox{\epsffile{h2.eps}}
\centerline{\rotu0}
\end{figure}">|; 

$key = q/f({{bf{v}),d^3{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1292.png"
 ALT="$f({\bf v})  d^3{\bf v}$">|; 

$key = q/{displaymath}int_{-infty}^{infty}exp(-beta,m,v_z^{~2}slash2),dv_z=sqrt{frac{2}{bty}^{infty}exp(-y^2),dy=sqrt{frac{2pi}{beta,m}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="494" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1288.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty} \exp(-\beta  m v_z^{&nbsp;2}/2)  dv_z ...
...fty}^{\infty} \exp(-y^2)  dy = \sqrt{\frac{2\pi}{\beta  m}},
\end{displaymath}">|; 

$key = q/n+dn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$n+dn$">|; 

$key = q/sum_rbar{n}_r=N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1411.png"
 ALT="$\sum_r\bar{n}_r = N$">|; 

$key = q/{displaymath}N_A=6times10^{23}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="\begin{displaymath}
N_A = 6 \times 10^{23}
\end{displaymath}">|; 

$key = q/{mitOmega}_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img700.png"
 ALT="${\mit\Omega}_s$">|; 

$key = q/c_V^{(e)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1585.png"
 ALT="$c_V^{(e)}$">|; 

$key = q/4,p^3,q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$4 p^3 q$">|; 

$key = q/{displaymath}frac{{mitDelta}^astn_1}{overline{n_1}}=sqrt{frac{q}{p}}frac{1}{sqrt{N}}ll1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="\begin{displaymath}
\frac{{\mit\Delta}^\ast n_1}{\overline{n_1}} = \sqrt{\frac{q}{p}}\frac{1}{\sqrt{N}}
\ll 1.
\end{displaymath}">|; 

$key = q/kappa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1084.png"
 ALT="$\kappa$">|; 

$key = q/{displaymath}beta,hbar,omega=frac{hbar,omega}{k,T}gg1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1104.png"
 ALT="\begin{displaymath}
\beta \hbar \omega = \frac{\hbar  \omega}{k T} \gg 1,
\end{displaymath}">|; 

$key = q/v_{{rm{rms};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$v_{\rm rms}$">|; 

$key = q/157^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$157^\circ $">|; 

$key = q/T_{{rm{rad};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1125.png"
 ALT="$T_{\rm rad}$">|; 

$key = q/L_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1202.png"
 ALT="$L_x$">|; 

$key = q/epsilonllmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1534.png"
 ALT="$\epsilon\ll \mu$">|; 

$key = q/{displaymath}bar{n}_s=N,frac{{{rm{e}^{-beta,epsilon_s}}{sum_r{{rm{e}^{-beta,epsilon_r}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1441.png"
 ALT="\begin{displaymath}
\bar{n}_s = N \frac{{\rm e}^{-\beta \epsilon_s}}
{\sum_r {\rm e}^{-\beta \epsilon_r}}.
\end{displaymath}">|; 

$key = q/T_1>T_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img888.png"
 ALT="$T_1 &gt; T_2$">|; 

$key = q/k_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1208.png"
 ALT="$k_z$">|; 

$key = q/x_alpha=tilde{x}_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img569.png"
 ALT="$x_\alpha=\tilde{x}_\alpha$">|; 

$key = q/Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="$Q$">|; 

$key = q/{displaymath}int_0^N{calP}(n),dnsimeq1{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="\begin{displaymath}
\int_0^N {\cal P}(n) dn \simeq 1
\end{displaymath}">|; 

$key = q/{mitOmega}_s=2^N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img702.png"
 ALT="${\mit\Omega}_s = 2^N$">|; 

$key = q/{mitOmega}(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="${\mit\Omega}(E)$">|; 

$key = q/{displaymath}E=frac{1}{2}sum_{i=1}^{3N}(p_i^{~2}+omega_i^{~2}q_i^{~2}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1178.png"
 ALT="\begin{displaymath}
E = \frac{1}{2}\sum_{i=1}^{3 N} (p_i^{&nbsp;2} + \omega_i^{&nbsp;2} q_i^{&nbsp;2}),
\end{displaymath}">|; 

$key = q/N=nu,N_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img727.png"
 ALT="$N= \nu  N_A$">|; 

$key = q/n=overline{n_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="$n= \overline{n_1}$">|; 

$key = q/{displaymath}Z=frac{exp[-(1slash2),beta,hbar,omega]}{1-exp(-beta,hbar,omega)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1095.png"
 ALT="\begin{displaymath}
Z = \frac{ \exp[-(1/2) \beta \hbar \omega]}{1-\exp(-\beta \hbar \omega)},
\end{displaymath}">|; 

$key = q/{displaymath}P_oplus=L_odot,frac{pi,R_oplus^{~2}slashr_oplus^{~2}}{4pi}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1515.png"
 ALT="\begin{displaymath}
P_\oplus=L_\odot \frac{ \pi  R_\oplus^{&nbsp;2}/r_\oplus^{&nbsp;2}}{4\pi}
\end{displaymath}">|; 

$key = q/-80^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img843.png"
 ALT="$-80^\circ$">|; 

$key = q/{displaymath}sigma=frac{3}{2}ln!left(frac{2,pi,m,k}{h_0^{~2}}right)+frac{3}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1041.png"
 ALT="\begin{displaymath}
\sigma = \frac{3}{2} \ln \!\left( \frac{2 \pi  m k}{h_0^{&nbsp;2}}\right) +
\frac{3}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{-q}{T_1}geq0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img874.png"
 ALT="\begin{displaymath}
\frac{-q}{T_1} \geq 0
\end{displaymath}">|; 

$key = q/h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="$h$">|; 

$key = q/g(u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$g(u)$">|; 

$key = q/{displaymath}d(lnZ+beta,overline{E})=beta(,{mathchar'26mskip-12mud}W+doverline{E})equivbeta,,{mathchar'26mskip-12mud}Q.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1004.png"
 ALT="\begin{displaymath}
d(\ln Z +\beta  \overline{E}) = \beta ( 
{\mathchar'26\msk...
... +d\overline{E}) \equiv \beta   {\mathchar'26\mskip-12mud}Q.
\end{displaymath}">|; 

$key = q/omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1186.png"
 ALT="$\omega$">|; 

$key = q/{displaymath}P(omega),domega=c,,overline{u}(omega),domega,frac{1}{2}!int_0^{pisleta,dtheta=frac{c}{4},overline{u}(omega),domega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1498.png"
 ALT="\begin{displaymath}
P(\omega)  d\omega = c   \overline{u}(\omega)  d\omega ...
...\theta d\theta = \frac{c}{4}  \overline{u}(\omega) d\omega,
\end{displaymath}">|; 

$key = q/{displaymath}rho=frac{p,mu}{R,T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img812.png"
 ALT="\begin{displaymath}
\rho = \frac{p \mu}{R T}.
\end{displaymath}">|; 

$key = q/{mitOmega}equiv{mitOmega}(E,x_1,cdots,x_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img576.png"
 ALT="${\mit\Omega}\equiv {\mit\Omega}(E, x_1,\cdots,
x_n)$">|; 

$key = q/n_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$n_1$">|; 

$key = q/{displaymath}overline{E}=nu,c_V,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1037.png"
 ALT="\begin{displaymath}
\overline{E} = \nu  c_V  T.
\end{displaymath}">|; 

$key = q/v^2,exp(-m,v^2slash2,k,T);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1340.png"
 ALT="$v^2 \exp(-m v^2/2 k T)$">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$\textstyle =$">|; 

$key = q/{displaymath}dE=left(frac{partialE}{partialT}right)_VdT.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img760.png"
 ALT="\begin{displaymath}
dE = \left(\frac{\partial E}{\partial T}\right)_V dT.
\end{displaymath}">|; 

$key = q/displaystylefrac{3}{4},frac{1}{(9pi)^{1slash3}},frac{m^2,c^3}{hbar}left(frac{M}{m_p}right)^{2slash3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1629.png"
 ALT="$\displaystyle \frac{3}{4} \frac{1}{(9\pi)^{1/3}} 
\frac{m^2 c^3}{\hbar} \left(\frac{M}{m_p}\right)^{2/3}.$">|; 

$key = q/{displaymath}P_{{rm{tot}(T)=frac{pi^2}{60}frac{k^4}{c^2,hbar^3},T^4=sigma,T^4.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1507.png"
 ALT="\begin{displaymath}
P_{\rm tot}(T) = \frac{\pi^2}{60} \frac{k^4}{c^2  \hbar^3}  T^4 = \sigma  T^4.
\end{displaymath}">|; 

$key = q/{displaymath}E+E'=E^{(0)}={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img561.png"
 ALT="\begin{displaymath}
E+E' = E^{(0)} = {\rm constant}.
\end{displaymath}">|; 

$key = q/displaystyleP(XmidY);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$\displaystyle P(X\mid Y)$">|; 

$key = q/k>k_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1554.png"
 ALT="$k&gt;k_F$">|; 

$key = q/n_s=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1394.png"
 ALT="$n_s=0$">|; 

$key = q/{displaymath}E=E(T,V).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img735.png"
 ALT="\begin{displaymath}
E= E(T, V).
\end{displaymath}">|; 

$key = q/{displaymath}1=sum_{n_1=0}^NP_N(n_1)simeqint_{-infty}^{infty}{calP}(n),dn{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="\begin{displaymath}
1= \sum_{n_1=0}^N P_N(n_1) \simeq \int_{-\infty}^{\infty}{\cal P}(n)  dn
\end{displaymath}">|; 

$key = q/d^3{{bf{k}equivdk_x,dk_y,dk_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1224.png"
 ALT="$d^3{\bf k}\equiv dk_x  dk_y  dk_z$">|; 

$key = q/{displaymath}c_p=c_V+R=frac{5}{2}R,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img786.png"
 ALT="\begin{displaymath}
c_p = c_V + R = \frac{5}{2} R,
\end{displaymath}">|; 

$key = q/{displaymath}frac{w}{T_1}geqq_2left(frac{1}{T_2}-frac{1}{T_1}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img914.png"
 ALT="\begin{displaymath}
\frac{w}{T_1} \geq q_2\left(\frac{1}{T_2} - \frac{1}{T_1}\right).
\end{displaymath}">|; 

$key = q/{displaymath}overline{E}simeqhbar,omega,[1slash2+exp(-beta,hbar,omega)]simeqfrac{1}{2},hbar,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1106.png"
 ALT="\begin{displaymath}
\overline{E} \simeq \hbar \omega  [ 1/2 + \exp(-\beta \hbar \omega)] \simeq
\frac{1}{2}  \hbar  \omega.
\end{displaymath}">|; 

$key = q/theta=QslashC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$\theta = Q/C$">|; 

$key = q/z<z_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img859.png"
 ALT="$z&lt;z_0$">|; 

$key = q/x_2,cdots,x_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="$x_2, \cdots, x_n$">|; 

$key = q/q_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="$q_z$">|; 

$key = q/T_B=T_C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img528.png"
 ALT="$T_B = T_C$">|; 

$key = q/bar{n}_sleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1415.png"
 ALT="$\bar{n}_s\leq 1$">|; 

$key = q/{displaymath}E_r=sum_{i=1}^Ne_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="\begin{displaymath}
E_r = \sum_{i=1}^N e_i.
\end{displaymath}">|; 

$key = q/{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1356.png"
 ALT="${\mit \Psi}$">|; 

$key = q/3!=3times2times1=6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$3!= 3\times 2 \times 1 = 6$">|; 

$key = q/E_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="$E_r$">|; 

$key = q/alpha_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1405.png"
 ALT="$\alpha_s$">|; 

$key = q/{displaymath}C,Vleft[int_{-infty}^{infty}exp(-beta,m,v_z^{~2}slash2),dv_zright]^3=N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1287.png"
 ALT="\begin{displaymath}
C  V \left[\int_{-\infty}^{\infty} \exp(-\beta  m v_z^{&nbsp;2}/2)  dv_z\right]^3
= N.
\end{displaymath}">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}G=tauleft(frac{partialsigma}{partialx},dx+frac{partialsigma}{partialy},dyright)=tau,dsigma,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}G = \tau \left( \frac{\partial \si...
... \frac{\partial\sigma}
{\partial y} dy\right)= \tau d\sigma,
\end{displaymath}">|; 

$key = q/n_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1148.png"
 ALT="$n_3$">|; 

$key = q/{displaymath}c_p=c_V+R{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img766.png"
 ALT="\begin{displaymath}
c_p = c_V + R
\end{displaymath}">|; 

$key = q/f({{bf{v});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1316.png"
 ALT="$f({\bf v})$">|; 

$key = q/{displaymath}frac{q}{T_1}=frac{w}{T_1}leq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img875.png"
 ALT="\begin{displaymath}
\frac{q}{T_1} = \frac{w}{T_1} \leq 0.
\end{displaymath}">|; 

$key = q/{displaymath}theta_E=frac{hbar,omega}{k}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1191.png"
 ALT="\begin{displaymath}
\theta_E = \frac{\hbar  \omega}{k}
\end{displaymath}">|; 

$key = q/k,Tllmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1582.png"
 ALT="$k T\ll \mu$">|; 

$key = q/{displaymath}V=6,pi^2,Nleft(frac{c_s}{omega_D}right)^3,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1251.png"
 ALT="\begin{displaymath}
V = 6 \pi^2  N \left(\frac{c_s}{\omega_D}\right)^3,
\end{displaymath}">|; 

$key = q/{displaymath}X_alpha=frac{1}{beta}frac{partialln{mitOmega}}{partialx_alpha}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img720.png"
 ALT="\begin{displaymath}
X_\alpha = \frac{1}{\beta} \frac{\partial \ln {\mit\Omega}}{\partial x_\alpha}
\end{displaymath}">|; 

$key = q/{displaymath}g(v_z),dv_z=int_{(v_x)}int_{(v_y)}f({{bf{v}),d^3{{bf{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1295.png"
 ALT="\begin{displaymath}
g(v_z) dv_z = \int_{(v_x)} \int_{(v_y)} f({\bf v}) d^3{\bf v}.
\end{displaymath}">|; 

$key = q/400^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img910.png"
 ALT="$400^\circ$">|; 

$key = q/E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img694.png"
 ALT="$E_0$">|; 

$key = q/p_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="$p_i$">|; 

$key = q/ln{mitOmega}(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="$\ln {\mit\Omega}(E)$">|; 

$key = q/{displaymath}U=int_0^{infty}overline{u}(omega),domega=frac{k,T}{pi^2,c^3}int_0^{infty}omega^2,domega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1475.png"
 ALT="\begin{displaymath}
U = \int_0^{\infty} \overline{u}(\omega)
 d\omega = \frac{k T}{\pi^2  c^3} \int_0^{\infty}
\omega^2 d\omega.
\end{displaymath}">|; 

$key = q/{displaymath}S'-2,S=2,nu,R,ln2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1049.png"
 ALT="\begin{displaymath}
S' - 2  S = 2 \nu  R  \ln 2.
\end{displaymath}">|; 

$key = q/{displaymath}bar{u},domega=epsilon(omega),sigma(omega),domega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1487.png"
 ALT="\begin{displaymath}
\bar{u} d\omega = \epsilon(\omega) \sigma(\omega) d\omega,
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}(E,V)proptoV^Nchi(E),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="\begin{displaymath}
{\mit\Omega}(E, V) \propto V^N \chi(E),
\end{displaymath}">|; 

$key = q/Tlltheta_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1260.png"
 ALT="$T\ll \theta_D$">|; 

$key = q/displaystyleS(T,V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1110.png"
 ALT="$\displaystyle S(T, V)$">|; 

$key = q/displaystyle1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1377.png"
 ALT="$\displaystyle 1,$">|; 

$key = q/dbar{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="$d \bar{E}$">|; 

$key = q/(3slash2),R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1149.png"
 ALT="$(3/2) R$">|; 

$key = q/partialoverline{E}slashpartialbetaleq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img987.png"
 ALT="$\partial \overline{E}/\partial \beta \leq 0$">|; 

$key = q/hbar,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1100.png"
 ALT="$\hbar  \omega$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W=frac{sum_rexp(-beta,E_r)(-partialE_rslashpartialx,,dx)}{sum_rexp(-beta,E_r)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img990.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W = \frac{ \sum_r \exp(-\beta  E_...
...-\partial E_r/\partial x   dx)}
{\sum_r \exp(-\beta  E_r)}.
\end{displaymath}">|; 

$key = q/{mitDelta}T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img751.png"
 ALT="${\mit\Delta} T$">|; 

$key = q/beta=1slashk,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img723.png"
 ALT="$\beta = 1/k T$">|; 

$key = q/N!slashN!=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$N!/N! = 1$">|; 

$key = q/Z_s(N-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1397.png"
 ALT="$Z_s(N-1)$">|; 

$key = q/displaystyleintexp[-(betaslash2m),p^2],d^3{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1026.png"
 ALT="$\displaystyle \int \exp[-(\beta/2m) p^2] d^3{\bf p}$">|; 

$key = q/{displaymath}sum_{i=1}^{M}P(X_i)=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="\begin{displaymath}
\sum_{i=1}^{M} P(X_i) =1,
\end{displaymath}">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1164.png"
 ALT="$J$">|; 

$key = q/{calP}(overline{n_1});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="${\cal P}(\overline{n_1})$">|; 

$key = q/L_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1204.png"
 ALT="$L_z$">|; 

$key = q/{displaymath}deltaE_r=frac{partialE_r}{partialx},dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img989.png"
 ALT="\begin{displaymath}
\delta E_r = \frac{\partial E_r}{\partial x}   dx.
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialln{mitOmega}(E_i)}{partialE},(bar{E}_f-E_i)+frac{partiamitOmega}'(E_i')}{partialE'},(bar{E}_f'-E_i')>0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img520.png"
 ALT="\begin{displaymath}
\frac{\partial\ln{\mit\Omega}(E_i)}{\partial E}  (\bar{E}_f...
...al\ln{\mit\Omega}'(E_i')}{\partial E'} (\bar{E}_f'-E_i') &gt; 0,
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}^astn_1=sqrt{N,p,q}gg1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="\begin{displaymath}
{\mit\Delta}^\ast n_1 = \sqrt{N p q}\gg 1,
\end{displaymath}">|; 

$key = q/E'+deltaE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="$E'+\delta E$">|; 

$key = q/{displaymath}v<|{{bf{v}|<v+dv.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1314.png"
 ALT="\begin{displaymath}
v &lt; \vert{\bf v}\vert &lt; v + dv.
\end{displaymath}">|; 

$key = q/epsilon_{-}=mu,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img946.png"
 ALT="$\epsilon_{-} = \mu   B$">|; 

$key = q/A-B<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1631.png"
 ALT="$A-B&lt;0$">|; 

$key = q/klnf;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img639.png"
 ALT="$k\ln f$">|; 

$key = q/{displaymath}p_F={mitLambda}left(frac{N}{V}right)^{1slash3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1597.png"
 ALT="\begin{displaymath}
p_F = {\mit\Lambda}\left(\frac{N}{V}\right)^{1/3},
\end{displaymath}">|; 

$key = q/{displaymath}frac{dP_r}{dt}=sum_{s}W_{rs}(P_s-P_r),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="\begin{displaymath}
\frac{dP_r}{dt} = \sum_{s} W_{rs}(P_s -P_r),
\end{displaymath}">|; 

$key = q/|A|^2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1359.png"
 ALT="$\vert A\vert^2=1$">|; 

$key = q/{displaymath}Z_s(N)=sum_{n_1,n_2,cdots}^{(s)}{{rm{e}^{-beta,(n_1,epsilon_1+n_2,epsilon_2+cdots)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img1393.png"
 ALT="\begin{displaymath}
Z_s(N) = \sum_{n_1,n_2,\cdots}^{(s)}
{\rm e}^{-\beta (n_1 \epsilon_1+n_2 \epsilon_2+
\cdots)},
\end{displaymath}">|; 

$key = q/{mathchar'26mskip-12mud}W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img714.png"
 ALT="${\mathchar'26\mskip-12mud}W$">|; 

$key = q/{displaymath}overline{(DeltaE)^2}=-frac{partialoverline{E}}{partialbeta}=frac{partial^2lnZ}{partialbeta^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img986.png"
 ALT="\begin{displaymath}
\overline{(\Delta E)^2} = -\frac{\partial \overline{E}}{\partial \beta}
= \frac{\partial^2 \ln Z}{\partial \beta^2}.
\end{displaymath}">|; 

$key = q/{displaymath}bar{u}equivsum_{i=1}^{M}P(u_i),u_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="\begin{displaymath}
\bar{u} \equiv \sum_{i=1}^{M} P(u_i)  u_i.
\end{displaymath}">|; 

$key = q/intd^3{{bf{r}_i=V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="$\int d^3{\bf r}_i= V$">|; 

$key = q/{displaymath}sigma_c(omega),domega=3,frac{k^2,V}{2pi^2},dk=3,frac{V}{2pi^2,c_s^{~3}},omega^2,domega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1230.png"
 ALT="\begin{displaymath}
\sigma_c(\omega) d\omega = 3  \frac{k^2 V}{2\pi^2}   dk = 3  \frac{V}
{2\pi^2  c_s^{&nbsp;3}}  \omega^2 d\omega.
\end{displaymath}">|; 

$key = q/beta,mull1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1531.png"
 ALT="$\beta \mu\ll 1$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W=overline{p},dV=frac{1}{beta}frac{partiallnZ}{partialV},dV{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img997.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W = \overline{p}  dV = \frac{1}{\beta} \frac{\partial \ln Z}{\partial V}
 dV
\end{displaymath}">|; 

$key = q/exp(-beta,E_r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img934.png"
 ALT="$\exp(-\beta  E_r)$">|; 

$key = q/rho,A,dz,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img806.png"
 ALT="$\rho A dz g$">|; 

$key = q/displaystylefrac{2pi}{L_z},n_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1214.png"
 ALT="$\displaystyle \frac{2\pi}{L_z}  n_z,$">|; 

$key = q/y=sqrt{beta},p_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1079.png"
 ALT="$ y = \sqrt{\beta}  p_i$">|; 

$key = q/2!=2times1=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$2! = 2\times 1 = 2$">|; 

$key = q/mu=mu_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1547.png"
 ALT="$\mu=\mu_0$">|; 

$key = q/deltaq;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$\delta q$">|; 

$key = q/{displaymath}n(omega)=frac{1}{{{rm{e}^{,beta,hbar,omega}-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1481.png"
 ALT="\begin{displaymath}
n(\omega) = \frac{1}{{\rm e}^{ \beta \hbar \omega}-1}.
\end{displaymath}">|; 

$key = q/H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$H$">|; 

$key = q/{displaymath}frac{V_b}{V_a}=frac{V_c}{V_d}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img902.png"
 ALT="\begin{displaymath}
\frac{ V_b}{V_a} = \frac{V_c}{V_d}.
\end{displaymath}">|; 

$key = q/{displaymath}N=sum_rn_r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1383.png"
 ALT="\begin{displaymath}
N = \sum_r n_r.
\end{displaymath}">|; 

$key = q/TllT_{{rm{rad};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1123.png"
 ALT="$T\ll T_{\rm rad}$">|; 

$key = q/{displaymath}{mitOmega}(y_1,cdots,y_n)rightarrow{{rm{maximum}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img614.png"
 ALT="\begin{displaymath}
{\mit\Omega}(y_1,\cdots, y_n)\rightarrow {\rm maximum}.
\end{displaymath}">|; 

$key = q/{displaymath}S^{(0)}=S(tilde{E})+S'(tilde{E}').{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img651.png"
 ALT="\begin{displaymath}
S^{(0)} = S(\tilde{E}) + S'(\tilde{E}').
\end{displaymath}">|; 

$key = q/{displaymath}T,V^{,gamma-1}={{rm{constant},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img799.png"
 ALT="\begin{displaymath}
T  V^{ \gamma-1} = {\rm constant},
\end{displaymath}">|; 

$key = q/deltaq_i,deltap_i=h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img692.png"
 ALT="$\delta q_i  \delta p_i = h$">|; 

$key = q/,{mathchar'26mskip-12mud}W=bar{p},dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img664.png"
 ALT="$ {\mathchar'26\mskip-12mud}W = \bar{p} dV$">|; 

$key = q/omega_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1179.png"
 ALT="$\omega_i$">|; 

$key = q/(p+q)^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$(p+q)^3$">|; 

$key = q/textstylesigma_c(omega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1236.png"
 ALT="$\textstyle \sigma_c(\omega)$">|; 

$key = q/6,400;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img825.png"
 ALT="$6,400$">|; 

$key = q/{displaymath}C_{n_1,,N-n_1}^N=frac{N!}{n_1!,(N-n_1)!}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="\begin{displaymath}
C_{n_1, N-n_1}^N = \frac{N!}{n_1 !  (N-n_1)!}.
\end{displaymath}">|; 

$key = q/{displaymath}w=nu,R,(T_1-T_2),lnfrac{V_b}{V_a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img904.png"
 ALT="\begin{displaymath}
w = \nu  R   (T_1 - T_2) \ln\frac{V_b}{V_a}.
\end{displaymath}">|; 

$key = q/0.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1170.png"
 ALT="$0.2$">|; 

$key = q/{mitOmega}'(E^{(0)}-E_r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img921.png"
 ALT="${\mit\Omega}'(E^{(0)} - E_r)$">|; 

$key = q/(partial{mitOmega}slashpartialx),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img547.png"
 ALT="$(\partial {\mit\Omega}/\partial x) dx$">|; 

$key = q/{mitOmega}gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img699.png"
 ALT="${\mit\Omega} \gg 1$">|; 

$key = q/V_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$V_f$">|; 

$key = q/{mitDelta}Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img750.png"
 ALT="${\mit\Delta} Q$">|; 

$key = q/{displaymath}C=frac{{mitDelta}Q}{{mitDelta}T}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img752.png"
 ALT="\begin{displaymath}
C = \frac{{\mit\Delta} Q}{{\mit\Delta} T}.
\end{displaymath}">|; 

$key = q/epsilon_{+}=-mu,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img944.png"
 ALT="$\epsilon_{+} = - \mu   B$">|; 

$key = q/gamma=1.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img863.png"
 ALT="$\gamma=1.4$">|; 

$key = q/{mathchar'26mskip-12mud}W=sum_alphaX_alphadx_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img722.png"
 ALT="${\mathchar'26\mskip-12mud}W = \sum_\alpha X_\alpha dx_\alpha$">|; 

$key = q/{displaymath}sum_{n_1=0}^NP_N(n_1)=sum_{n_1=0}^Nfrac{N!}{n_1!,(N-n_1)!},p^{n_1},q^{N-n_1}equiv(p+q)^N=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="\begin{displaymath}
\sum_{n_1=0}^N P_N(n_1) =\sum_{n_1=0}^N \frac{N!}{n_1! (N-n_1)!} p^{n_1} 
q^{N-n_1}\equiv (p+q)^N = 1,
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}n_x=frac{L_x}{2pi},dk_x.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1219.png"
 ALT="\begin{displaymath}
{\mit\Delta} n_x = \frac{L_x}{2\pi}  dk_x.
\end{displaymath}">|; 

$key = q/R(E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="$R(E)$">|; 

$key = q/R(E+deltaE);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="$R(E+\delta E)$">|; 

$key = q/1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$1/2$">|; 

$key = q/z_1=[gammaslash(gamma-1)],z_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img862.png"
 ALT="$z_1 = [\gamma/(\gamma-1)] z_0$">|; 

$key = q/bar{E}=tilde{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img571.png"
 ALT="$\bar{E}=\tilde{E}$">|; 

$key = q/displaystyle(N,p)^2+N,p,q=(overline{n_1})^2+N,p,q,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$\displaystyle (N p)^2 + N p q = (\overline{n_1})^2 + N p q,$">|; 

$key = q/beta,(epsilon-mu)ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1535.png"
 ALT="$\beta (\epsilon-\mu)\ll 1$">|; 

$key = q/{displaymath}W_{if}=int_{V_i}^{V_f}{mathchar'26mskip-12mud}W=int_{V_i}^{V_f}bar{p}(V),dV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="\begin{displaymath}
W_{if} = \int_{V_i}^{V_f} {\mathchar'26\mskip-12mud}W = \int_{V_i}^{V_f} \bar{p}(V)  dV.
\end{displaymath}">|; 

$key = q/{displaymath}Z=frac{zeta^N}{N!}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1055.png"
 ALT="\begin{displaymath}
Z = \frac{\zeta^N}{N!}.
\end{displaymath}">|; 

$key = q/omegaslashc_s+domegaslashc_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1229.png"
 ALT="$\omega/c_s + d\omega/c_s$">|; 

$key = q/1slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$1/3$">|; 

$key = q/{displaymath}frac{{mitDelta}^astE}{bar{E}}simfrac{1}{sqrt{f}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="\begin{displaymath}
\frac{{\mit\Delta}^\ast E}{\bar{E}}\sim \frac{1}{\sqrt{f}}.
\end{displaymath}">|; 

$key = q/{displaymath}f_D(y)rightarrowfrac{4pi^4}{5}frac{1}{y^3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1262.png"
 ALT="\begin{displaymath}
f_D(y) \rightarrow \frac{4\pi^4}{5} \frac{1}{y^3}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{{mitOmega}_f}{{mitOmega}_i}simeq10^{,2,nutimes10^{23}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img618.png"
 ALT="\begin{displaymath}
\frac{{\mit\Omega}_f}{{\mit\Omega}_i} \simeq 10^{ 2 \nu\times 10^{23}},
\end{displaymath}">|; 

$key = q/6times5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$6\times 5$">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img771.png"
 ALT="$\rho$">|; 

$key = q/,{mathchar'26mskip-12mud}W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="$ {\mathchar'26\mskip-12mud}W$">|; 

$key = q/{displaymath}int_0^yfrac{expx}{(expx-1)^2},x^4,dxsimeqint_0^inftyfrac{expx}{(expx-1)^2},x^4,dx=frac{4pi^4}{15}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1261.png"
 ALT="\begin{displaymath}
\int_0^y \frac{\exp x}{(\exp x -1)^2} x^4   dx \simeq
\int...
...y \frac{\exp x}{(\exp x -1)^2} x^4   dx = \frac{4\pi^4}{15}.
\end{displaymath}">|; 

$key = q/1slash36;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$1/36$">|; 

$key = q/P_r=P_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="$P_r=P_s$">|; 

$key = q/{displaymath}{mitDelta}^astu=left[overline{({mitDelta}u)^2}right]^{1slash2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="\begin{displaymath}
{\mit\Delta}^\ast u = \left[\overline{({\mit\Delta} u)^2}\right]^{1/2},
\end{displaymath}">|; 

$key = q/{displaymath}E_{rs}^{(0)}=E_r+E_s'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1009.png"
 ALT="\begin{displaymath}
E_{rs}^{(0)} = E_r +E_s'.
\end{displaymath}">|; 

$key = q/{displaymath}c_Vsimeqfrac{12pi^4}{5}Rleft(frac{T}{theta_D}right)^3{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1263.png"
 ALT="\begin{displaymath}
c_V \simeq \frac{12 \pi^4}{5} R \left(\frac{T}{\theta_D}\right)^3
\end{displaymath}">|; 

$key = q/dS;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img588.png"
 ALT="$dS$">|; 

$key = q/{mitDelta}E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1118.png"
 ALT="${\mit\Delta} E$">|; 

$key = q/{displaymath}overline{E}^{(0)}=overline{E}+overline{E}'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1018.png"
 ALT="\begin{displaymath}
\overline{E}^{(0)} = \overline{E} +\overline{E}'.
\end{displaymath}">|; 

$key = q/displaystyleleft[sum_rexp(-beta,E_r)right]left[sum_sexp(-beta,E_s')right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1014.png"
 ALT="$\displaystyle \left[\sum_r \exp(-\beta  E_r)\right]\left[\sum_s
\exp(-\beta  E_s')\right].$">|; 

$key = q/{displaymath}{mitOmega}^{(0)}(E)={mitOmega}(E),{mitOmega}'(E'),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img642.png"
 ALT="\begin{displaymath}
{\mit\Omega}^{(0)}(E) = {\mit\Omega}(E)  {\mit\Omega}'(E'),
\end{displaymath}">|; 

$key = q/epsilon_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1380.png"
 ALT="$\epsilon_r$">|; 

$key = q/overline{(n_1)^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$\overline{(n_1)^2}$">|; 

$key = q/{displaymath}bar{n}_s=frac{sum_Rn_s,exp[-beta,(n_1,epsilon_1+n_2,epsilon_2+cdotsexp[-beta,(n_1,epsilon_1+n_2,epsilon_2+cdots)]}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1387.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{\sum_R n_s \exp[-\beta (n_1 \epsilon_1+...
...sum_R \exp[-\beta (n_1 \epsilon_1+n_2 \epsilon_2+\cdots)]}.
\end{displaymath}">|; 

$key = q/{displaymath}P_i=frac{{mitOmega}_i}{{mitOmega}_f}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img598.png"
 ALT="\begin{displaymath}
P_i =\frac{{\mit\Omega}_i}{{\mit\Omega}_f}
\end{displaymath}">|; 

$key = q/1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$1$">|; 

$key = q/N_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img728.png"
 ALT="$N_A$">|; 

$key = q/C_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1568.png"
 ALT="$C_V$">|; 

$key = q/T=T';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img661.png"
 ALT="$T=T'$">|; 

$key = q/{displaymath}c_V=-frac{3,N_A,hbar,omega}{k,T^2}left[-frac{exp(beta,hbar,omega),hbar,omega}{[exp(beta,hbar,omega)-1]^2}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1189.png"
 ALT="\begin{displaymath}
c_V = - \frac{3  N_A  \hbar  \omega}{k  T^2} \left[
-\fr...
...\hbar  \omega}
{[\exp(\beta \hbar  \omega) - 1]^2} \right],
\end{displaymath}">|; 

$key = q/{displaymath}overline{u}(omega),domega=k,T,sigma(omega),domega=frac{k,T}{pi^2,c^3},omega^2,domega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1474.png"
 ALT="\begin{displaymath}
\overline{u}(\omega) d\omega = k T  \sigma(\omega)  d\omega = \frac{k T}
{\pi^2  c^3}
 \omega^2 d\omega.
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{3,V}{{mitLambda}^3}int_0^{p_F}(p^2,c^2+m^2,c^4)^{1slash2},p^2,dp,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1622.png"
 ALT="\begin{displaymath}
K = \frac{3 V}{{\mit\Lambda}^3}\int_0^{p_F} (p^2 c^2
+m^2 c^4)^{1/2} p^2 dp,
\end{displaymath}">|; 

$key = q/{displaymath}overline{v}=4pileft(frac{m}{2pi,k,T}right)^{3slash2}int_0^inftyv^3exp(-m,v^2slash2,k,T),dv,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1329.png"
 ALT="\begin{displaymath}
\overline{v} = 4\pi \left(\frac{m}{2\pi  k T}\right)^{3/2}
\int_0^\infty v^3 \exp
(-m v^2/2 k T) dv,
\end{displaymath}">|; 

$key = q/{displaymath}ln{mitOmega}(bar{E}_f)+ln{mitOmega}'(bar{E}_f')>ln{mitOmega}(E_i)+ln{mitOmega}'(E_i'),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img519.png"
 ALT="\begin{displaymath}
\ln{\mit\Omega}(\bar{E}_f) + \ln{\mit\Omega}'(\bar{E}_f')&gt;\ln{\mit\Omega}(E_i) + \ln{\mit\Omega}'(E_i'),
\end{displaymath}">|; 

$key = q/bar{n}_srightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1413.png"
 ALT="$\bar{n}_s\rightarrow 0$">|; 

$key = q/overline{mu}_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img940.png"
 ALT="$\overline{\mu}_B$">|; 

$key = q/15^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img844.png"
 ALT="$15^\circ$">|; 

$key = q/{displaymath}{calP}(overline{n_1})simeqfrac{1}{sqrt{2pi},{mitDelta}^astn_1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="\begin{displaymath}
{\cal P}(\overline{n_1})\simeq \frac{1}{
\sqrt{2\pi}  {\mit\Delta}^\ast n_1}.
\end{displaymath}">|; 

$key = q/{mitOmega}'(E^{(0)}-E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="${\mit\Omega}'(E^{(0)}-E)$">|; 

$key = q/displaystylefrac{1}{T}left(frac{partialE}{partialV}right)_T+frac{nu,R}{V}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img745.png"
 ALT="$\displaystyle \frac{1}{T} \left(\frac{\partial E}
{\partial V} \right)_T + \frac{\nu  R}{V}.$">|; 

$key = q/displaystylebar{E}_f-E_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img513.png"
 ALT="$\displaystyle \bar{E}_f - E_i,$">|; 

$key = q/cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1371.png"
 ALT="$\cdots$">|; 

$key = q/y_alpha+dy_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img594.png"
 ALT="$y_\alpha+d y_\alpha$">|; 

$key = q/{displaymath}sum_rn_r=N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1392.png"
 ALT="\begin{displaymath}
\sum_r n_r = N.
\end{displaymath}">|; 

$key = q/{mitDelta}N=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1409.png"
 ALT="${\mit\Delta} N =1$">|; 

$key = q/{displaymath}sum_rfrac{1}{{{rm{e}^{,alpha+beta,epsilon_r}-1}=N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1433.png"
 ALT="\begin{displaymath}
\sum_r
\frac{1}{{\rm e}^{ \alpha+\beta \epsilon_r}-1} =N.
\end{displaymath}">|; 

$key = q/{displaymath}overline{E}=-frac{1}{Z}frac{partialZ}{partialbeta}=-frac{partiallnZ}{partialbeta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img979.png"
 ALT="\begin{displaymath}
\overline{E} = - \frac{1}{Z} \frac{\partial Z}{\partial \beta} = - \frac{
\partial \ln Z}{\partial \beta}.
\end{displaymath}">|; 

$key = q/displaystylepleft[N,(p+q)^{N-1}+p,N,(N-1),(p+q)^{N-2}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle p\left[N (p+q)^{N-1}+p N (N-1) (p+q)^{N-2}\right].$">|; 

$key = q/{displaymath}Z=intcdotsintexp(-beta,E),frac{d^3{{bf{r}_1cdotsd^3{{bf{r}_N,d^3{{bf{p}_1cdotsd^3{{bf{p}_N}{h_0^{~3N}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="412" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1022.png"
 ALT="\begin{displaymath}
Z = \int\cdots \int \exp(-\beta   E)  \frac{d^3{\bf r}_1\cdots d^3{\bf r}_N 
d^3{\bf p}_1\cdots d^3{\bf p}_N}{h_0^{&nbsp;3N}},
\end{displaymath}">|; 

$key = q/{mathchar'26mskip-12mud}Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img713.png"
 ALT="${\mathchar'26\mskip-12mud}Q$">|; 

$key = q/{displaymath}theta_1,C_1+theta_2,C_2=theta,C_1+theta,C_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="\begin{displaymath}
\theta_1  C_1 + \theta_2  C_2 = \theta  C_1 + \theta  C_2,
\end{displaymath}">|; 

$key = q/theta+dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1492.png"
 ALT="$\theta + d\theta$">|; 

$key = q/10^{13};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="$10^{13}$">|; 

$key = q/{displaymath}omega=k,c_s.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1200.png"
 ALT="\begin{displaymath}
\omega = k  c_s.
\end{displaymath}">|; 

$key = q/Y=partialFslashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="$Y=\partial F/\partial y$">|; 

$key = q/{displaymath}P_{+}=Cexp(-beta,epsilon_{+})=Cexp(beta,mu,B),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img947.png"
 ALT="\begin{displaymath}
P_{+} = C\exp(-\beta \epsilon_{+}) = C\exp(\beta  \mu  B),
\end{displaymath}">|; 

$key = q/cdots,x_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="$\cdots, x_n$">|; 

$key = q/{displaymath}E=frac{3}{2},nu,R,T,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img782.png"
 ALT="\begin{displaymath}
E = \frac{3}{2}  \nu  R T,
\end{displaymath}">|; 

$key = q/{displaymath}S^{(0)}=S+S'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1019.png"
 ALT="\begin{displaymath}
S^{(0)} = S + S'.
\end{displaymath}">|; 

$key = q/k,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img533.png"
 ALT="$k T$">|; 

$key = q/H_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$H_1$">|; 

$key = q/1+x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1258.png"
 ALT="$1+x$">|; 

$key = q/{displaymath}sum_rbar{n}_r=sum_rfrac{1}{{{rm{e}^{,beta,(epsilon_r-mu)}+1}=N,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1527.png"
 ALT="\begin{displaymath}
\sum_r \bar{n}_r = \sum_r\frac{1}{{\rm e}^{ \beta (\epsilon_r-\mu)} + 1}
=N,
\end{displaymath}">|; 

$key = q/{displaymath}overline{E}simeqhbar,omegaleft[frac{1}{2}+frac{1}{beta,hbar,omega}reqhbar,omegaleft[frac{1}{beta,hbar,omega}right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1102.png"
 ALT="\begin{displaymath}
\overline{E} \simeq \hbar \omega\left[\frac{1}{2} + \frac{1...
...meq \hbar \omega\left[ \frac{1}{\beta \hbar \omega}\right],
\end{displaymath}">|; 

$key = q/-60^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img845.png"
 ALT="$-60^\circ$">|; 

$key = q/{displaymath}omega=sqrt{frac{kappa}{m}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1091.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{\kappa}{m}}.
\end{displaymath}">|; 

$key = q/sigma_c(omega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1233.png"
 ALT="$\sigma_c(\omega)$">|; 

$key = q/P_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$P_s$">|; 

$key = q/x+dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="$x+dx$">|; 

$key = q/Z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img980.png"
 ALT="$Z$">|; 

$key = q/{displaymath}w=q_1-q_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img903.png"
 ALT="\begin{displaymath}
w = q_1 - q_2,
\end{displaymath}">|; 

$key = q/otimes;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$\otimes$">|; 

$key = q/displaystylefrac{1}{2}kappa,overline{x^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1087.png"
 ALT="$\displaystyle \frac{1}{2}\kappa  \overline{x^2}$">|; 

$key = q/{{bf{p}=hbar,{{bf{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1548.png"
 ALT="${\bf p} =\hbar {\bf k}$">|; 

$key = q/n(omega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1491.png"
 ALT="$n(\omega)$">|; 

$key = q/1slash6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$1/6$">|; 

$key = q/{displaymath}lnint_{-infty}^{infty}exp(-beta,epsilon_i),dp_i=-frac{1}{2}lnbeta+lnint_{-infty}^{infty}exp(-b,y^2),dy.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1080.png"
 ALT="\begin{displaymath}
\ln \int_{-\infty}^{\infty} \exp(-\beta  \epsilon_i)  dp_i...
... \ln \beta
+ \ln \int_{-\infty}^{\infty} \exp(- b  y^2)  dy.
\end{displaymath}">|; 

$key = q/{displaymath}c_p=frac{1}{nu}left(frac{{mathchar'26mskip-12mud}Q}{dT}right)_p,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img765.png"
 ALT="\begin{displaymath}
c_p = \frac{1}{\nu}\left(\frac{{\mathchar'26\mskip-12mud}Q}{dT}\right)_p,
\end{displaymath}">|; 

$key = q/{displaymath}frac{|P_N(n_1+1)-P_N(n_1)|}{P_N(n_1)}ll1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="\begin{displaymath}
\frac{\vert P_N(n_1+1)-P_N(n_1)\vert}{P_N(n_1)} \ll 1.
\end{displaymath}">|; 

$key = q/5times10^9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1140.png"
 ALT="$5\times 10^9$">|; 

$key = q/{mitDelta}^astEslashbar{E}sim10^{-12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="${\mit\Delta}^\ast E /\bar{E}\sim 10^{-12}$">|; 

$key = q/c_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1201.png"
 ALT="$c_s$">|; 

$key = q/{displaymath}dS=left(frac{partialS}{partialT}right)_VdT+left(frac{partialS}{partialV}right)_TdV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img740.png"
 ALT="\begin{displaymath}
dS = \left(\frac{\partial S}{\partial T}\right)_V dT + \left( \frac{\partial S}
{\partial V}\right)_T dV.
\end{displaymath}">|; 

$key = q/{displaymath}overline{({mitDelta}n_1)^2}=overline{(n_1)^2}-(overline{n_1})^2=N,p,q.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="\begin{displaymath}
\overline{({\mit\Delta} n_1)^2}= \overline{(n_1)^2}- (\overline{n_1})^2 = N p q.
\end{displaymath}">|; 

$key = q/{displaymath}S_f-S_i=int_i^fdS=int_i^ffrac{{mathchar'26mskip-12mud}Q}{T},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img589.png"
 ALT="\begin{displaymath}
S_f - S_i = \int_i^f dS = \int_i^f \frac{{\mathchar'26\mskip-12mud}Q}{T},
\end{displaymath}">|; 

$key = q/deltaRproptodeltaEslashE^{,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$\delta R \propto \delta E/ E^{ 1/2}$">|; 

$key = q/displaystyleQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img512.png"
 ALT="$\displaystyle Q$">|; 

$key = q/{displaymath}sum_rbar{n}_r=sum_rfrac{1}{{{rm{e}^{,alpha+beta,epsilon_r}pm1}=N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1443.png"
 ALT="\begin{displaymath}
\sum_r \bar{n}_r = \sum_r \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_r} \pm 1}
=N.
\end{displaymath}">|; 

$key = q/Ssimkf;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img638.png"
 ALT="$S\sim kf$">|; 

$key = q/373^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img907.png"
 ALT="$373^\circ$">|; 

$key = q/F(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="$F(x, y)$">|; 

$key = q/{displaymath}lnZ=alpha,Npmsum_rleft(pm{{rm{e}^{-alpha-beta,epsilon_r}right)=alpha,N+N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1453.png"
 ALT="\begin{displaymath}
\ln Z = \alpha N \pm \sum_r\left(\pm {\rm e}^{-\alpha-\beta \epsilon_r}\right)
=\alpha N +N.
\end{displaymath}">|; 

$key = q/overline{n}=overline{n_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$\overline{n}=\overline{n_1}$">|; 

$key = q/s_1,cdots,s_N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1355.png"
 ALT="$s_1,\cdots,s_N$">|; 

$key = q/c,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1497.png"
 ALT="$c \cos\theta$">|; 

$key = q/Srightarrow-infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1042.png"
 ALT="$S\rightarrow -\infty$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{kcv.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="557" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img1591.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{kcv.eps}}
\end{figure}">|; 

$key = q/displaystyle{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1478.png"
 ALT="$\displaystyle {\bf p}$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}W_r=-dE_r=sum_{alpha=1}^nX_{alpha,r},dx_alpha,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}W_r = - d E_r = \sum_{\alpha=1}^n X_{\alpha r} dx_\alpha,
\end{displaymath}">|; 

$key = q/displaystyleE-tilde{E},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="$\displaystyle E - \tilde{E},$">|; 

$key = q/E_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img504.png"
 ALT="$E_i$">|; 

$key = q/displaystyleP(X),P(Y).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$\displaystyle P(X) P(Y).$">|; 

$key = q/{figure}{epsfysize{=3.5in{setbox{0=hbox{{epsffile{diamond.eps}{}{centerline{rotu0}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="406" BORDER="0"
 SRC="|."$dir".q|img1245.png"
 ALT="\begin{figure}
\epsfysize =3.5in\setbox 0=\hbox{\epsffile{diamond.eps}}
\centerline{\rotu0}
\end{figure}">|; 

$key = q/displaystylexi_{{rm{MB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1374.png"
 ALT="$\displaystyle \xi_{\rm MB}$">|; 

$key = q/{displaymath}B_1=-lntilde{n}+ln,(N-tilde{n})+lnp-lnq.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="\begin{displaymath}
B_1 = -\ln\tilde{n} +\ln (N-\tilde{n})+\ln p - \ln q.
\end{displaymath}">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=dbar{E},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = d \bar{E},
\end{displaymath}">|; 

$key = q/1slashsqrt{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$1/\sqrt{N}$">|; 

$key = q/{displaymath}frac{dH}{dt}=sum_rleft(frac{dP_r}{dt}lnP_r+frac{dP_r}{dt}right)=sum_rfrac{dP_r}{dt},(lnP_r+1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="\begin{displaymath}
\frac{dH}{dt} = \sum_r \left(\frac{d P_r}{dt} \ln P_r +\frac{d P_r}
{dt}\right) = \sum_r \frac{d P_r}{dt} (\ln P_r + 1).
\end{displaymath}">|; 

$key = q/{displaymath}bar{X}_alphaequiv-overline{frac{partialE_r}{partialx_alpha}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="\begin{displaymath}
\bar{X}_\alpha \equiv -\overline{ \frac{\partial E_r}{\partial x_\alpha}}
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}F=F_f-F_i=int_i^fdF=int_i^f(X,dx+Y,dy).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="\begin{displaymath}
{\mit\Delta} F = F_f - F_i = \int_i^f dF= \int_i^f(X dx + Y dy).
\end{displaymath}">|; 

$key = q/v_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1284.png"
 ALT="$v_x$">|; 

$key = q/{displaymath}h=frac{h_1,A_1+h_2,A_2}{A_1+A_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="\begin{displaymath}
h = \frac{h_1  A_1 + h_2  A_2}{A_1 + A_2}.
\end{displaymath}">|; 

$key = q/1slashN_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img623.png"
 ALT="$1/N_A$">|; 

$key = q/P(u_2),cdots,P(u_M);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$P(u_2),\cdots, P(u_M)$">|; 

$key = q/F(epsilon)=exp[-beta,(epsilon-mu)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1538.png"
 ALT="$F(\epsilon)=
\exp[-\beta (\epsilon-\mu)]$">|; 

$key = q/0.02,theta_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1265.png"
 ALT="$0.02  \theta_D$">|; 

$key = q/{displaymath}oint{mathchar'26mskip-12mud}Gneq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="\begin{displaymath}
\oint {\mathchar'26\mskip-12mud}G \neq 0.
\end{displaymath}">|; 

$key = q/{{bf{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="${\bf x}$">|; 

$key = q/{displaymath}sum_r{{rm{e}^{-alpha-beta,epsilon_r}=N.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1452.png"
 ALT="\begin{displaymath}
\sum_r {\rm e}^{-\alpha-\beta \epsilon_r} = N.
\end{displaymath}">|; 

$key = q/x_1,cdots,x_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img555.png"
 ALT="$x_1, \cdots, x_n$">|; 

$key = q/4^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img916.png"
 ALT="$4^\circ$">|; 

$key = q/{displaymath}P_r=Cexp(-beta,E_r),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img931.png"
 ALT="\begin{displaymath}
P_r = C \exp(-\beta  E_r),
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}proptoV^N,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img602.png"
 ALT="\begin{displaymath}
{\mit\Omega} \propto V^N,
\end{displaymath}">|; 

$key = q/{displaymath}W_{rs}=W_{sr}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="\begin{displaymath}
W_{rs} = W_{sr}
\end{displaymath}">|; 

$key = q/{displaymath}left(frac{partial}{partialV}right)_Tleft(frac{partialS}{partialT}riT}right)_Vleft(frac{partialS}{partialV}right)_T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img747.png"
 ALT="\begin{displaymath}
\left(\frac{\partial}{\partial V}\right)_T \left(\frac{\part...
...ial T}\right)_V \left(\frac{\partial S}
{\partial V}\right)_T.
\end{displaymath}">|; 

$key = q/{displaymath}overline{({mitDelta}n_1)^2}equiv({mitDelta}^astn_1)^2=sum_{n_1=0}^N-infty}^{infty}{calP}(n),(n-overline{n_1})^2,dn,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="\begin{displaymath}
\overline{({\mit\Delta} n_1)^2}\equiv
({\mit\Delta}^\ast n_1...
...\int_{-\infty}^{\infty}{\cal P}(n)
 (n-\overline{n_1})^2 dn,
\end{displaymath}">|; 

$key = q/dnrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$dn\rightarrow 0$">|; 

$key = q/gamma=c_pslashc_V=(c_V+R)slashc_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1173.png"
 ALT="$\gamma = c_p / c_V
= (c_V + R)/ c_V$">|; 

$key = q/{displaymath}overline{{mitDelta}u}=overline{(u-bar{u})}=bar{u}-bar{u}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="\begin{displaymath}
\overline{{\mit\Delta} u} = \overline{(u-\bar{u})} = \bar{u}-\bar{u} = 0.
\end{displaymath}">|; 

$key = q/x_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="$x_\alpha$">|; 

$key = q/{displaymath}frac{1}{nu}left(frac{partial[NE_0]}{partialT}right)_V=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1116.png"
 ALT="\begin{displaymath}
\frac{1}{\nu}\left(
\frac{\partial [N E_0]}{\partial T}\right)_V = 0.
\end{displaymath}">|; 

$key = q/z=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img816.png"
 ALT="$z=0$">|; 

$key = q/10^9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1128.png"
 ALT="$10^9$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1311.png"
 ALT="$v$">|; 

$key = q/{displaymath}{mitLambda}=(3pi^2)^{1slash3},hbar.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1598.png"
 ALT="\begin{displaymath}
{\mit\Lambda} = (3\pi^2)^{1/3} \hbar.
\end{displaymath}">|; 

$key = q/{displaymath}c_V=3R,f_D(beta,hbar,omega_D)=3,R,f_D(theta_DslashT),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1252.png"
 ALT="\begin{displaymath}
c_V = 3R f_D(\beta  \hbar \omega_D)= 3 R f_D(\theta_D/T),
\end{displaymath}">|; 

$key = q/{mitSigma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="${\mit\Sigma}$">|; 

$key = q/exp(-beta,epsilon^{{rm{int}_s);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1276.png"
 ALT="$\exp(-\beta  \epsilon^{\rm int}_s)$">|; 

$key = q/overline{n_1}=N,p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$\overline{n_1}= N p$">|; 

$key = q/{{bf{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1199.png"
 ALT="${\bf k}$">|; 

$key = q/100^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img906.png"
 ALT="$100^\circ$">|; 

$key = q/{displaymath}h_0geqhbar.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="\begin{displaymath}
h_0\geq \hbar.
\end{displaymath}">|; 

$key = q/{displaymath}frac{R_ast}{R_odot}=0.010left(frac{M_odot}{M}right)^{1slash3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1616.png"
 ALT="\begin{displaymath}
\frac{R_\ast}{R_\odot}= 0.010\left(\frac{M_\odot}{M}\right)^{1/3},
\end{displaymath}">|; 

$key = q/p_i^{~2}={{bf{p}_i!cdot!{{bf{p}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1021.png"
 ALT="$p_i^{&nbsp;2} = {\bf p}_i\!\cdot \!{\bf p}_i$">|; 

$key = q/displaystyle0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$\displaystyle 0,$">|; 

$key = q/I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1163.png"
 ALT="$I$">|; 

$key = q/8.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img822.png"
 ALT="$8.4$">|; 

$key = q/{displaymath}k=frac{R}{N_A}~~~{{rm{joulesslashkelvin},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img624.png"
 ALT="\begin{displaymath}
k = \frac{R}{N_A}&nbsp;&nbsp;&nbsp;{\rm joules/kelvin},
\end{displaymath}">|; 

$key = q/{displaymath}sum_rP_r(t)=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="\begin{displaymath}
\sum_r P_r(t) =1.
\end{displaymath}">|; 

$key = q/{displaymath}Z=sum_{n_1,n_2,cdots}frac{N!}{n_1!,n_2!,cdots}({{rm{e}^{-beta,epsil})^{n_1},({{rm{e}^{-beta,epsilon_2})^{n_2}cdots,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1438.png"
 ALT="\begin{displaymath}
Z = \sum_{n_1,n_2,\cdots} \frac{N!}{n_1 ! n_2! \cdots}
({...
...psilon_1})^{n_1} ({\rm e}^{-\beta \epsilon_2})^{n_2}
\cdots,
\end{displaymath}">|; 

$key = q/{displaymath}S^ast=S+klnfrac{delta^astE}{deltaE}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img636.png"
 ALT="\begin{displaymath}
S^\ast = S + k \ln\frac{\delta^\ast E}{\delta E}.
\end{displaymath}">|; 

$key = q/T_A=T_C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img529.png"
 ALT="$T_A=T_C$">|; 

$key = q/k_z+dk_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1221.png"
 ALT="$k_z + d k_z $">|; 

$key = q/{displaymath}beta=frac{partialln{mitOmega}}{partialE},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img524.png"
 ALT="\begin{displaymath}
\beta =\frac{\partial \ln{\mit\Omega}}{\partial E},
\end{displaymath}">|; 

$key = q/n_{i3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="$n_{i3}$">|; 

$key = q/14,432;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img826.png"
 ALT="$14,432$">|; 

$key = q/bar{x}_alpha=tilde{x}_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img572.png"
 ALT="$\bar{x}_\alpha= \tilde{x}_\alpha$">|; 

$key = q/{displaymath}overline{E}=frac{1}{2},k,T+frac{1}{2},k,T=k,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1089.png"
 ALT="\begin{displaymath}
\overline{E} = \frac{1}{2}  k T + \frac{1}{2}  k T = k T.
\end{displaymath}">|; 

$key = q/{bf{p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1268.png"
 ALT="$\bf p$">|; 

$key = q/iequiv(x_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="$i\equiv (x_i$">|; 

$key = q/{displaymath}dln{mitOmega}=betaleft(dbar{E}+sum_alphabar{X}_alpha,dbar{x}_alpharight).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img580.png"
 ALT="\begin{displaymath}
d\ln{\mit\Omega} = \beta\left(d\bar{E} + \sum_\alpha \bar{X}_\alpha d\bar{x}_\alpha
\right).
\end{displaymath}">|; 

$key = q/{displaymath}beta_f=beta_f',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img508.png"
 ALT="\begin{displaymath}
\beta_f = \beta_f',
\end{displaymath}">|; 

$key = q/dn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$dn$">|; 

$key = q/Srightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img705.png"
 ALT="$S\rightarrow 0$">|; 

$key = q/{displaymath}overline{E}=-frac{partiallnZ}{partialbeta}=frac{3}{2}frac{N}{beta}=nu,frac{3}{2},R,T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1035.png"
 ALT="\begin{displaymath}
\overline{E} = - \frac{\partial \ln Z}{\partial \beta} = \frac{3}{2} \frac{N}{\beta}
= \nu  \frac{3}{2}  R  T.
\end{displaymath}">|; 

$key = q/{displaymath}overline{f(u)}equivsum_{i=1}^{M}P(u_i),f(u_i).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="\begin{displaymath}
\overline{f(u)} \equiv \sum_{i=1}^{M} P(u_i)  f(u_i).
\end{displaymath}">|; 

$key = q/{displaymath}{mitOmega}proptoE^f,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="\begin{displaymath}
{\mit\Omega} \propto E^f,
\end{displaymath}">|; 

$key = q/s_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1353.png"
 ALT="$s_j$">|; 

$key = q/{displaymath}p+q=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="\begin{displaymath}
p+q=1,
\end{displaymath}">|; 

$key = q/{displaymath}ln{mitOmega}_f-ln{mitOmega}_i=nu,N_A,ln2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img621.png"
 ALT="\begin{displaymath}
\ln {\mit\Omega}_f - \ln {\mit\Omega}_i = \nu  N_A \ln 2,
\end{displaymath}">|; 

$key = q/M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$M$">|; 

$key = q/P(E_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img506.png"
 ALT="$P(E_i)$">|; 

$key = q/{displaymath}S_f-S_iequiv{mitDelta}Sgeq0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img629.png"
 ALT="\begin{displaymath}
S_f - S_i \equiv {\mit\Delta} S \geq 0,
\end{displaymath}">|; 

$key = q/{displaymath}bar{n}_s=frac{1}{{{rm{e}^{,beta,(epsilon_s-mu)}+1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1525.png"
 ALT="\begin{displaymath}
\bar{n}_s = \frac{1}{{\rm e}^{ \beta (\epsilon_s-\mu)} + 1},
\end{displaymath}">|; 

$key = q/{displaymath}lnP_r=lnC'+ln{mitOmega}'(E^{(0)})-beta,E_r,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img930.png"
 ALT="\begin{displaymath}
\ln P_r = \ln C' + \ln {\mit\Omega}'(E^{(0)}) - \beta  E_r,
\end{displaymath}">|; 

$key = q/(1slash2),N,k,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1112.png"
 ALT="$(1/2) N k T$">|; 

$key = q/displaystyleleft(frac{partial^2S}{partialV^2}right)_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img683.png"
 ALT="$\displaystyle \left(\frac{\partial^2 S}{\partial V^2}\right)_E$">|; 

$key = q/{displaymath}p,V^{,gamma}={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img797.png"
 ALT="\begin{displaymath}
p  V^{ \gamma} = {\rm constant}.
\end{displaymath}">|; 

$key = q/rho({{bf{k}),d^3{{bf{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1468.png"
 ALT="$\rho({\bf k}) 
d^3{\bf k}$">|; 

$key = q/T_A=T_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img527.png"
 ALT="$T_A= T_B$">|; 

$key = q/u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$u_1$">|; 

$key = q/(3slash2),k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1577.png"
 ALT="$(3/2) k$">|; 

$key = q/v_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1285.png"
 ALT="$v_y$">|; 

$key = q/{displaymath}h_1,A_1+h_2,A_2=h,A_1+h,A_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="\begin{displaymath}
h_1  A_1 + h_2  A_2 = h  A_1 + h  A_2,
\end{displaymath}">|; 

$key = q/c_VproptoT^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1196.png"
 ALT="$c_V \propto T^3$">|; 

$key = q/{displaymath}P_isimexpleft(-10^{24}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img606.png"
 ALT="\begin{displaymath}
P_i \sim \exp\left(-10^{24}\right).
\end{displaymath}">|; 

$key = q/epsilon_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1414.png"
 ALT="$\epsilon_s$">|; 

$key = q/U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1594.png"
 ALT="$U$">|; 

$key = q/{displaymath}int_{-infty}^inftyexp(-beta,epsilon_i),epsilon_i,dp_iequiv-frac{part_{-infty}^inftyexp(-beta,epsilon_i),dp_iright],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="410" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1076.png"
 ALT="\begin{displaymath}
\int_{-\infty}^\infty
\exp(-\beta  \epsilon_i) \epsilon_i...
...\int_{-\infty}^\infty \exp(-\beta  \epsilon_i)  dp_i\right],
\end{displaymath}">|; 

$key = q/{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1279.png"
 ALT="${\bf v}$">|; 

$key = q/5times10^{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1171.png"
 ALT="$5\times 10^{12}$">|; 

$key = q/{displaymath}bar{epsilon}(omega)=hbar,omega,,n(omega)=frac{hbar,omega}{{{rm{e}^{,beta,hbar,omega}-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1482.png"
 ALT="\begin{displaymath}
\bar{\epsilon}(\omega) = \hbar \omega  n(\omega) = \frac{\hbar \omega}{{\rm e}^{ \beta \hbar \omega}-1}.
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{infty}exp(-x^2),dx=sqrt{pi},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{\infty}
\exp(-x^2) dx = \sqrt{\pi},
\end{displaymath}">|; 

$key = q/f(u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$f(u)$">|; 

$key = q/10^{24};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$10^{24}$">|; 

$key = q/fequiv(x_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$f \equiv (x_f$">|; 

$key = q/{displaymath}frac{R_ast}{R_odot}=0.000011left(frac{M_odot}{M}right)^{1slash3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1638.png"
 ALT="\begin{displaymath}
\frac{R_\ast}{R_\odot}= 0.000011\left(\frac{M_\odot}{M}\right)^{1/3},
\end{displaymath}">|; 

$key = q/{displaymath}k,Tsimfrac{bar{E}}{f},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img532.png"
 ALT="\begin{displaymath}
k T \sim \frac{\bar{E}}{f},
\end{displaymath}">|; 

$key = q/betaequiv1slashk,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img581.png"
 ALT="$\beta\equiv 1/k T$">|; 

$key = q/{displaymath}(n_1)^2,p^{n_1}equivleft(p,frac{partial}{partialp}right)^2p^{n_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="\begin{displaymath}
(n_1)^2  p^{n_1} \equiv \left(p \frac{\partial}{\partial p}\right)^2 p^{n_1},
\end{displaymath}">|; 

$key = q/{displaymath}c_V=frac{g}{2},R,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1115.png"
 ALT="\begin{displaymath}
c_V = \frac{g}{2}  R,
\end{displaymath}">|; 

$key = q/bar{E}_f';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img507.png"
 ALT="$\bar{E}_f'$">|; 

$key = q/n_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1147.png"
 ALT="$n_2$">|; 

$key = q/{displaymath}overline{n_1}=sum_{n_1=0}^NP_N(n_1),n_1simeqint_{-infty}^{infty}{calP}(n),n,dn,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="\begin{displaymath}
\overline{n_1} = \sum_{n_1=0}^N P_N(n_1) n_1 \simeq \int_{-\infty}^{\infty}
{\cal P}(n) n dn,
\end{displaymath}">|; 

$key = q/h,nu=k,T_{{rm{rad};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1120.png"
 ALT="$h  \nu = k  T_{\rm rad}$">|; 

$key = q/{displaymath}M<M_C,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1633.png"
 ALT="\begin{displaymath}
M&lt; M_C,
\end{displaymath}">|; 

$key = q/{displaymath}frac{{mitDelta}^astE}{overline{E}}simfrac{1}{sqrt{f}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img709.png"
 ALT="\begin{displaymath}
\frac{{\mit\Delta}^\ast E}{\overline{E}} \sim \frac{1}{\sqrt{f}},
\end{displaymath}">|; 

$key = q/{displaymath}omega=k,c{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1465.png"
 ALT="\begin{displaymath}
\omega = k  c
\end{displaymath}">|; 

$key = q/{displaymath}dN=frac{3,V}{{mitLambda}^3},p^2,dp.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1601.png"
 ALT="\begin{displaymath}
dN = \frac{3 V}{{\mit\Lambda}^3}  p^2 dp.
\end{displaymath}">|; 

$key = q/0.241;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="$0.241$">|; 

$key = q/{displaymath}Z=sum_{n_1,n_2,cdots}frac{N!}{n_1!,n_2!,cdots}{{rm{e}^{-beta,(n_1,epsilon_1+n_2,epsilon_2+cdots)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1437.png"
 ALT="\begin{displaymath}
Z = \sum_{n_1,n_2,\cdots} \frac{N!}{n_1 ! n_2! \cdots} {\rm e}^{-\beta (n_1 \epsilon_1+n_2 \epsilon_2+\cdots)},
\end{displaymath}">|; 

$key = q/{displaymath}overline{(DeltaE)^2}=overline{E^2}-overline{E}^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img981.png"
 ALT="\begin{displaymath}
\overline{(\Delta E)^2} = \overline{E^2} - \overline{E}^2
\end{displaymath}">|; 

$key = q/{displaymath}P_s({{bf{r},{{bf{p}),d^3{{bf{r},d^3{{bf{p}proptoexp(-beta,p^2slash2eta,epsilon^{{rm{int}_s),,d^3{{bf{r},d^3{{bf{p},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1274.png"
 ALT="\begin{displaymath}
P_s({\bf r}, {\bf p})  d^3{\bf r} d^3{\bf p} \propto
\exp...
... \exp(-\beta \epsilon^{\rm int}_s)  d^3{\bf r} d^3{\bf p},
\end{displaymath}">|; 

$key = q/nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img619.png"
 ALT="$\nu$">|; 

$key = q/(VslashN)^{1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1563.png"
 ALT="$(V/N)^{1/3}$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=nu,c_V,dT+nuR,dT.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img764.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = \nu  c_V   dT + \nu R  dT.
\end{displaymath}">|; 

$key = q/tilde{n}simoverline{n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$\tilde{n}\sim
\overline{n}$">|; 

$key = q/v^3,exp(-m,v^2slash2,k,T);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1339.png"
 ALT="$v^3 \exp(-m v^2/2 k T)$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Qequivdbar{E}+{mathchar'26mskip-12mud}W{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q \equiv d \bar{E} + {\mathchar'26\mskip-12mud}W
\end{displaymath}">|; 

$key = q/7slash5=1.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1176.png"
 ALT="$7/5 = 1.4$">|; 

$key = q/{{bf{p}+d{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1272.png"
 ALT="${\bf p} + d{\bf p}$">|; 

$key = q/displaystylexi_{{rm{BE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1376.png"
 ALT="$\displaystyle \xi_{\rm BE}$">|; 

$key = q/{displaymath}bar{X}(E,x)=-overline{frac{partialE_r}{partialx}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img546.png"
 ALT="\begin{displaymath}
\bar{X}(E,x) = -\overline{\frac{\partial E_r}{\partial x}}
\end{displaymath}">|; 

$key = q/5slash3=1.667;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1174.png"
 ALT="$5/3 = 1.667$">|; 

$key = q/{displaymath}overline{v}=frac{1}{n}int_0^inftyF(v),v,dv.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1328.png"
 ALT="\begin{displaymath}
\overline{v} = \frac{1}{n} \int_0^\infty F(v) v dv.
\end{displaymath}">|; 

$key = q/y+dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="$y+dy$">|; 

$key = q/displaystyleQ_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1350.png"
 ALT="$\displaystyle Q_j,$">|; 

$key = q/,{mathchar'26mskip-12mud}W=sumbar{X}_alpha,dx_alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="$ {\mathchar'26\mskip-12mud}W = \sum \bar{X}_\alpha dx_\alpha$">|; 

$key = q/{displaymath}lambda_Fequivfrac{2pi}{k_F}=frac{2pi}{(3pi^2)^{1slash3}}left(frac{V}{N}right)^{1slash3},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1561.png"
 ALT="\begin{displaymath}
\lambda_F \equiv \frac{2\pi}{k_F} =\frac{2\pi}{(3\pi^2)^{1/3}}
\left(\frac{V}{N}\right)^{1/3},
\end{displaymath}">|; 

$key = q/displaystyleoverline{(n_1)^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$\displaystyle \overline{(n_1)^2}$">|; 

$key = q/theta_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1254.png"
 ALT="$\theta_D$">|; 

$key = q/{displaymath}c_V=frac{1}{nu}left(frac{partialoverline{E}}{partialT}right)_V=frac{3}{2},R,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1036.png"
 ALT="\begin{displaymath}
c_V = \frac{1}{\nu} \left(\frac{\partial \overline{E}}{\partial T} \right)_V
= \frac{3}{2}  R,
\end{displaymath}">|; 

$key = q/{displaymath}E_r+E'=E^{(0)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img919.png"
 ALT="\begin{displaymath}
E_r + E' = E^{(0)},
\end{displaymath}">|; 

$key = q/dF;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="$dF$">|; 

$key = q/lnZ_s(N-{mitDelta}N);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1400.png"
 ALT="$\ln Z_s(N-{\mit\Delta}N)$">|; 

$key = q/{displaymath}overline{n_1}=N,pgg1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="\begin{displaymath}
\overline{n_1} = N p \gg 1.
\end{displaymath}">|; 

$key = q/{displaymath}overline{epsilon}_i=-frac{partial}{partialbeta}left(-frac{1}{2}lnbetaright)=frac{1}{2,beta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1081.png"
 ALT="\begin{displaymath}
\overline{\epsilon}_i = - \frac{\partial}{\partial \beta} \left( - \frac{1}{2}
\ln \beta\right) = \frac{1}{2 \beta},
\end{displaymath}">|; 

$key = q/{displaymath}frac{dP_r}{dt}=sum_{sneqr}P_sW_{sr}-sum_{sneqr}P_rW_{rs},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="\begin{displaymath}
\frac{dP_r}{dt} = \sum_{s\neq r} P_s W_{sr} - \sum_{s\neq r} P_r W_{rs},
\end{displaymath}">|; 

$key = q/p^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$p^4$">|; 

$key = q/10^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img817.png"
 ALT="$10^5$">|; 

$key = q/P(n,n+dn);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$P(n, n+dn)$">|; 

$key = q/{displaymath}overline{v}=sqrt{frac{8}{pi}frac{k,T}{m}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1332.png"
 ALT="\begin{displaymath}
\overline{v} = \sqrt{\frac{8}{\pi} \frac{k T}{m}}.
\end{displaymath}">|; 

$key = q/displaystyleC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1628.png"
 ALT="$\displaystyle C$">|; 

$key = q/Q=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$Q=0$">|; 

$key = q/{displaymath}{mathchar'26mskip-12mud}Q=nu,c_V,dT+p,dV=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img789.png"
 ALT="\begin{displaymath}
{\mathchar'26\mskip-12mud}Q = \nu  c_V  dT +p dV = 0,
\end{displaymath}">|; 

$key = q/(p+q)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$(p+q)^2$">|; 

$key = q/{displaymath}F(v),dv=4pi,nleft(frac{m}{2pi,k,T}right)^{3slash2}v^2,exp(-m,v^2slash2,k,T),dv.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1321.png"
 ALT="\begin{displaymath}
F(v) dv = 4\pi  n\left(\frac{m}{2\pi  k T}\right)^{3/2} v^2 \exp(-m v^2/2 k T) dv.
\end{displaymath}">|; 

$key = q/{displaymath}p,V={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img788.png"
 ALT="\begin{displaymath}
p V = {\rm constant}.
\end{displaymath}">|; 

$key = q/displaystyleleft(frac{partialS}{partialE}right)_V,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img671.png"
 ALT="$\displaystyle \left(\frac{\partial S}{\partial E}\right)_V,$">|; 

$key = q/O^{16};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1605.png"
 ALT="$O^{16}$">|; 

$key = q/bar{E}_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="$\bar{E}_f$">|; 

$key = q/mrightarrowm_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1637.png"
 ALT="$m\rightarrow m_p$">|; 

$key = q/k<k_F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1553.png"
 ALT="$k&lt;k_F$">|; 

$key = q/{displaymath}S=k,(lnZ+beta,overline{E}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1038.png"
 ALT="\begin{displaymath}
S = k (\ln Z + \beta  \overline{E}).
\end{displaymath}">|; 

$key = q/{displaymath}bar{n}_s=frac{1}{[Z_s(N)slashZ_s(N-1)],{{rm{e}^{,beta,epsilon_s}+1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1396.png"
 ALT="\begin{displaymath}
\bar{n}_s =\frac{1}{[Z_s(N)/Z_s(N-1)] {\rm e}^{ \beta \epsilon_s} + 1}.
\end{displaymath}">|; 

$key = q/x_1',cdots,x_n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img560.png"
 ALT="$x_1',\cdots, x_n'$">|; 

1;

