Fermi-Dirac statistics

Let us, first of all, consider Fermi-Dirac statistics. According to Eq. (582), the average number of particles in quantum state $s$ can be written

$$\bar{n}_s = \frac{\sum_{n_s} n_s\,{\rm e}^{-\beta\,n_s\,\eps... ... {\rm e}^{-\beta\,(n_1\,\epsilon_1+n_2\,\epsilon_2+ \cdots)}}.$$ (584)

Here, we have rearranged the order of summation, using the multiplicative properties of the exponential function. Note that the first sums in the numerator and denominator only involve $n_s$, whereas the last sums omit the particular state $s$ from consideration (this is indicated by the superscript $s$ on the summation symbol). Of course, the sums in the above expression range over all values of the numbers $n_1, n_2,\cdots$ such that $n_r=0$ and 1 for each $r$, subject to the overall constraint that

$$\sum_r n_r = N.$$ (585)

Let us introduce the function

$$Z_s(N) = \sum_{n_1,n_2,\cdots}^{(s)} {\rm e}^{-\beta\,(n_1\,\epsilon_1+n_2\,\epsilon_2+ \cdots)},$$ (586)

which is defined as the partition function for $N$ particles distributed over all quantum states, excluding state $s$, according to Fermi-Dirac statistics. By explicitly performing the sum over $n_s=0$ and 1, the expression (584) reduces to

$$\bar{n}_s = \frac{0 + {\rm e}^{-\beta\,\epsilon_s}\,Z_s(N-1)} {Z_s(N) + {\rm e}^{-\beta\,\epsilon_s}\,Z_s(N-1)},$$ (587)

which yields

$$\bar{n}_s =\frac{1}{[Z_s(N)/Z_s(N-1)]\,{\rm e}^{\,\beta\,\epsilon_s} + 1}.$$ (588)

In order to make further progress, we must somehow relate $Z_s(N-1)$ to $Z_s(N)$. Suppose that ${\mit\Delta}N\ll N$. It follows that $\ln Z_s(N-{\mit\Delta}N)$ can be Taylor expanded to give

$$\ln Z_s(N-{\mit\Delta}N) \simeq \ln Z_s(N) - \frac{\partial ... ...al N}\, {\mit\Delta} N = \ln Z_s(N) - \alpha_s\,{\mit\Delta}N,$$ (589)

where

$$\alpha_s\equiv \frac{\partial \ln Z_s}{\partial N}.$$ (590)

As always, we Taylor expand the slowly varying function $\ln Z_s(N)$, rather than the rapidly varying function $Z_s(N)$, because the radius of convergence of the latter Taylor series is too small for the series to be of any practical use. Equation (589) can be rearranged to give

$$Z_s(N-{\mit\Delta}N) = Z_s(N)\,{\rm e}^{-\alpha_s\,{\mit\Delta}N}.$$ (591)

Now, since $Z_s(N)$ is a sum over very many different quantum states, we would not expect the logarithm of this function to be sensitive to which particular state $s$ is excluded from consideration. Let us, therefore, introduce the approximation that $\alpha_s$ is independent of $s$, so that we can write

$$\alpha_s \simeq \alpha$$ (592)

for all $s$. It follows that the derivative (590) can be expressed approximately in terms of the derivative of the full partition function $Z(N)$ (in which the $N$ particles are distributed over all quantum states). In fact,

$$\alpha\simeq \frac{\partial \ln Z}{\partial N}.$$ (593)

Making use of Eq. (591), with ${\mit\Delta} N =1$, plus the approximation (592), the expression (588) reduces to

$$\bar{n}_s = \frac{1}{{\rm e}^{\,\alpha+\beta\,\epsilon_s}+ 1}.$$ (594)

This is called the Fermi-Dirac distribution. The parameter $\alpha$ is determined by the constraint that $\sum_r\bar{n}_r = N$: i.e.,

$$\sum_r \frac{1}{{\rm e}^{\,\alpha+\beta\,\epsilon_r}+ 1} = N.$$ (595)

Note that $\bar{n}_s\rightarrow 0$ if $\epsilon_s$ becomes sufficiently large. On the other hand, since the denominator in Eq. (594) can never become less than unity, no matter how small $\epsilon_s$ becomes, it follows that $\bar{n}_s\leq 1$. Thus,

$$0 \leq \bar{n}_s \leq 1,$$ (596)

in accordance with the Pauli exclusion principle.

Equations (583) and (593) can be integrated to give

$$\ln Z = \alpha\,N + \sum_r\ln \,(1+{\rm e}^{-\alpha-\beta\,\epsilon_r}),$$ (597)

where use has been made of Eq. (594).