Maxwell-Boltzmann statistics

For the purpose of comparison, it is instructive to consider the purely classical case of Maxwell-Boltzmann statistics. The partition function is written

$$Z=\sum_{R} {\rm e}^{-\beta\,(n_1\,\epsilon_1+n_2\,\epsilon_2+\cdots)},$$ (609)

where the sum is over all distinct states $R$ of the gas, and the particles are treated as distinguishable. For given values of $n_1, n_2,\cdots$ there are

$$\frac{N!}{n_1 !\,n_2!\,\cdots}$$ (610)

possible ways in which $N$ distinguishable particles can be put into individual quantum states such that there are $n_1$ particles in state 1, $n_2$ particles in state 2, etc. Each of these possible arrangements corresponds to a distinct state for the whole gas. Hence, Eq. (609) can be written

$$Z = \sum_{n_1,n_2,\cdots} \frac{N!}{n_1 !\,n_2!\,\cdots} {\rm e}^{-\beta\,(n_1\,\epsilon_1+n_2\,\epsilon_2+\cdots)},$$ (611)

where the sum is over all values of $n_r=0,1,2,\cdots$ for each $r$, subject to the constraint that

$$\sum_r n_r = N.$$ (612)

Now, Eq. (611) can be written

$$Z = \sum_{n_1,n_2,\cdots} \frac{N!}{n_1 !\,n_2!\,\cdots} ({... ...psilon_1})^{n_1}\,({\rm e}^{-\beta\,\epsilon_2})^{n_2} \cdots,$$ (613)

which, by virtue of Eq. (612), is just the result of expanding a polynomial. In fact,

$$Z = ({\rm e}^{-\beta\,\epsilon_1}+{\rm e}^{-\beta\,\epsilon_2}+ \cdots)^N,$$ (614)

or

$$\ln Z = N\,\ln\!\left(\sum_r {\rm e}^{-\beta\,\epsilon_r}\right).$$ (615)

Note that the argument of the logarithm is simply the partition function for a single particle.

Equations (583) and (615) can be combined to give

$$\bar{n}_s = N\,\frac{{\rm e}^{-\beta\,\epsilon_s}} {\sum_r {\rm e}^{-\beta\,\epsilon_r}}.$$ (616)

This is known as the Maxwell-Boltzmann distribution. It is, of course, just the result obtained by applying the Boltzmann distribution to a single particle (see Sect. 7).