Fermi-Dirac Statistics

Let us, first of all, consider Fermi-Dirac statistics. According to Equation (8.19), the average number of particles in quantum state $s$ can be written

$$\bar{n}_s = \frac{\sum_{n_s} n_s {\rm e}^{-\beta n_s \epsilon_... ...n_2,\cdots}^{(s)} {\rm e}^{-\beta (n_1 \epsilon_1+n_2 \epsilon_2+ \cdots)}}.$$ (8.21)

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 previous 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

$$\sum_r n_r = N.$$ (8.22)

Let us introduce the function

$$Z_s(N) = \sum_{n_1,n_2,\cdots}^{(s)} {\rm e}^{-\beta (n_1 \epsilon_1+n_2 \epsilon_2+ \cdots)},$$ (8.23)

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 (8.21) reduces to

$$\bar{n}_s = \frac{0 + {\rm e}^{-\beta \epsilon_s} Z_s(N-1)} {Z_s(N) + {\rm e}^{-\beta \epsilon_s} Z_s(N-1)},$$ (8.24)

which yields

$$\bar{n}_s =\frac{1}{[Z_s(N)/Z_s(N-1)] {\rm e}^{ \beta \epsilon_s} + 1}.$$ (8.25)

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

$$\ln Z_s(N-{\mit\Delta}N) \simeq \ln Z_s(N) - \frac{\partial \ln Z_s}{\partial N} {\mit\Delta} N = \ln Z_s(N) - \alpha_s {\mit\Delta}N,$$ (8.26)

where

$$\alpha_s\equiv \frac{\partial \ln Z_s}{\partial N}.$$ (8.27)

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 (8.26) can be rearranged to give

$$Z_s(N-{\mit\Delta}N) = Z_s(N) {\rm e}^{-\alpha_s {\mit\Delta}N}.$$ (8.28)

Now, because $Z_s(N)$ 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

$$\alpha_s \simeq \alpha$$ (8.29)

for all $s$ . It follows that the derivative (8.27) 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,

$$\alpha\simeq \frac{\partial \ln Z}{\partial N}.$$ (8.30)

Making use of Equation (8.28), with ${\mit\Delta} N =1$ , plus the approximation (8.29), the expression (8.25) reduces to

$$\bar{n}_s = \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_s}+ 1}.$$ (8.31)

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

$$\sum_r \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_r}+ 1} = N.$$ (8.32)

Note that $\bar{n}_s\rightarrow 0$ if $\epsilon_s$ becomes sufficiently large. On the other hand, because the denominator in Equation (8.31) can never become less than unity, no matter how small $\epsilon_s$ becomes, it follows that $\bar{n}_s\leq 1$ . Thus,

$$0 \leq \bar{n}_s \leq 1,$$ (8.33)

in accordance with the Pauli exclusion principle.

Equations (8.20) and (8.30) can be integrated to give

$$\ln Z = \alpha N + \sum_r\ln\left(1+{\rm e}^{-\alpha-\beta \epsilon_r}\right),$$ (8.34)

where use has been made of Equation (8.31).