Exercises

Consider the Maxwellian distribution

$\displaystyle f ({\bf v})= n\left(\frac{m}{2\pi\,T}\right)^{3/2}\,\exp\left[-\frac{m\,({\bf v}-{\bf V})^{2}}{2\,T}\right].$
Demonstrate that

$\displaystyle n$ $\displaystyle = \int f\,d^3{\bf v},$

$\displaystyle n\,{\bf V}$ $\displaystyle = \int {\bf v}\,f\,d^3{\bf v},$

$\displaystyle \frac{3}{2}\,n\,T$ $\displaystyle = \int \frac{1}{2}\,m\,v^2\,f\,d^3{\bf v}.$
The species- entropy per unit volume is conventionally defined as

$\displaystyle s_s= -\int f_s\,\ln f_s\,d^3{\bf v}_s.$
The Boltzmann H-theorem thus states that collisions drive the system toward a maximum entropy state characterized by Maxwellian distribution functions with common mean velocities and common temperatures. Demonstrate that for a Maxwellian distribution,

$\displaystyle f_s= n_s\left(\frac{m_s}{2\pi\,T_s}\right)^{3/2}\,\exp\left(-\frac{m\,v_s^{2}}{2\,T_s}\right),$
the entropy per unit volume takes the form

$\displaystyle s_s= n_s\left[\ln\left(\frac{T_s^{3/2}}{n_s}\right) +\frac{3}{2}\ln\left(\frac{2\pi}{m_s}\right)+\frac{3}{2}\right].$
The Landau collision operator is written

$\displaystyle C_{ss'}(f_s,f_{s'}) = \frac{\gamma_{ss'}}{m_s}\,\frac{\partial}{\partial{\bf v}_s}\cdot\int {\bf w}_{ss'}\cdot {\bf J}_{ss'}\,d^3{\bf v}_{s'},$
where

$\displaystyle \gamma_{ss'}$ $\displaystyle = \left(\frac{e_s\,e_{s'}}{4\pi\,\epsilon_0}\right)^2 2\pi\,\ln{\mit\Lambda}_c,$

$\displaystyle {\bf w}_{ss'}$ $\displaystyle = \frac{u_{ss'}^{2}\,{\bf I} - {\bf u}_{ss'}{\bf u}_{ss'}}{u_{ss'}^{3}},$

$\displaystyle u_{ss'}$ $\displaystyle = \vert{\bf v}_s-{\bf v}_{s'}\vert,$

$\displaystyle {\bf J}_{ss'}$ $\displaystyle = \frac{\partial f_s}{\partial{\bf v}_s}\,\frac{f_{s'}}{m_s} -\frac{f_s}{m_{s'}}\,\frac{\partial f_{s'}}{\partial {\bf v}_{s'}}.$

Demonstrate directly that this collision operator satisfies the same conservation laws as the Boltzmann collision operator. Namely,

$\displaystyle \int C_{ss'}\,d^3{\bf v}_s$ $\displaystyle =0,$

$\displaystyle \int m_s\,{\bf v}_s\,C_{ss'}\,d^3{\bf v}_s$ $\displaystyle = -\int m_{s'}\,{\bf v}_{s'}\,C_{s's}\,d^3{\bf v}_{s'},$

$\displaystyle \int\frac{1}{2}\,m_s\,v_s^{2}\,C_{ss'}\,d^3{\bf v}_s$ $\displaystyle =- \int\frac{1}{2}\,m_{s'}\,v_{s'}^{2}\,C_{s's}\,d^3{\bf v}_{s'}.$

$\displaystyle n$	$\displaystyle = \int f\,d^3{\bf v},$
$\displaystyle n\,{\bf V}$	$\displaystyle = \int {\bf v}\,f\,d^3{\bf v},$
$\displaystyle \frac{3}{2}\,n\,T$	$\displaystyle = \int \frac{1}{2}\,m\,v^2\,f\,d^3{\bf v}.$

$\displaystyle \gamma_{ss'}$	$\displaystyle = \left(\frac{e_s\,e_{s'}}{4\pi\,\epsilon_0}\right)^2 2\pi\,\ln{\mit\Lambda}_c,$
$\displaystyle {\bf w}_{ss'}$	$\displaystyle = \frac{u_{ss'}^{2}\,{\bf I} - {\bf u}_{ss'}{\bf u}_{ss'}}{u_{ss'}^{3}},$
$\displaystyle u_{ss'}$	$\displaystyle = \vert{\bf v}_s-{\bf v}_{s'}\vert,$
$\displaystyle {\bf J}_{ss'}$	$\displaystyle = \frac{\partial f_s}{\partial{\bf v}_s}\,\frac{f_{s'}}{m_s} -\frac{f_s}{m_{s'}}\,\frac{\partial f_{s'}}{\partial {\bf v}_{s'}}.$

$\displaystyle \int C_{ss'}\,d^3{\bf v}_s$	$\displaystyle =0,$
$\displaystyle \int m_s\,{\bf v}_s\,C_{ss'}\,d^3{\bf v}_s$	$\displaystyle = -\int m_{s'}\,{\bf v}_{s'}\,C_{s's}\,d^3{\bf v}_{s'},$
$\displaystyle \int\frac{1}{2}\,m_s\,v_s^{2}\,C_{ss'}\,d^3{\bf v}_s$	$\displaystyle =- \int\frac{1}{2}\,m_{s'}\,v_{s'}^{2}\,C_{s's}\,d^3{\bf v}_{s'}.$