Bose-Einstein statistics

Let us now consider Bose-Einstein statistics. The particles in the system are assumed to be massive, so the total number of particles $N$ is a fixed number.

Consider the expression (584). For the case of massive bosons, the numbers $n_1, n_2,\cdots$ assume all values $n_r=0,1,2,\cdots$ for each $r$, subject to the constraint that $\sum_r n_r=N$. Performing explicitly the sum over $n_s$, this expression reduces to

$$\bar{n}_s=\frac{0 +{\rm e}^{-\beta\,\epsilon_s}\,Z_s(N-1) + ... ...,Z_s(N-1) + {\rm e}^{-2\,\beta\,\epsilon_s}\,Z_s(N-2)+\cdots},$$ (604)

where $Z_s(N)$ is the partition function for $N$ particles distributed over all quantum states, excluding state $s$, according to Bose-Einstein statistics [cf., Eq. (586)]. Using Eq. (591), and the approximation (592), the above equation reduces to

$$\bar{n}_s = \frac{\sum_s n_s\,{\rm e}^{-n_s\,(\alpha+\beta\,... ...ilon_s)}} {\sum_s {\rm e}^{-n_s\,(\alpha+\beta\,\epsilon_s)}}.$$ (605)

Note that this expression is identical to (598), except that $\beta\,\epsilon_s$ is replaced by $\alpha+\beta\,\epsilon_s$. Hence, an analogous calculation to that outlined in the previous subsection yields

$$\bar{n}_s = \frac{1}{{\rm e}^{\,\alpha+\beta\,\epsilon_s}-1}.$$ (606)

This is called the Bose-Einstein distribution. Note that $\bar{n}_s$ can become very large in this distribution. The parameter $\alpha$ is again determined by the constraint on the total number of particles: i.e.,

$$\sum_r \frac{1}{{\rm e}^{\,\alpha+\beta\,\epsilon_r}-1} =N.$$ (607)

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

$$\ln Z = \alpha\,N - \sum_r\ln \,(1-{\rm e}^{-\alpha-\beta\,\epsilon_r}),$$ (608)

where use has been made of Eq. (606).

Note that photon statistics correspond to the special case of Bose-Einstein statistics in which the parameter $\alpha$ takes the value zero, and the constraint (607) does not apply.