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 (8.21). 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

$$\bar{n}_s=\frac{0 +{\rm e}^{-\beta \epsilon_s} Z_s(N-1) + 2 {\... ...eta \epsilon_s} Z_s(N-1) + {\rm e}^{-2 \beta \epsilon_s} Z_s(N-2)+\cdots},$$ (8.41)

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., Equation (8.23)]. Using Equation (8.28), and the approximation (8.29), the previous equation reduces to

$$\bar{n}_s = \frac{\sum_s n_s {\rm e}^{-n_s (\alpha+\beta \epsilon_s)}} {\sum_s {\rm e}^{-n_s (\alpha+\beta \epsilon_s)}}.$$ (8.42)

Note that this expression is identical to (8.35), except that $\beta \epsilon_s$ is replaced by $\alpha+\beta \epsilon_s$ . Hence, an analogous calculation to that outlined in the previous section yields

$$\bar{n}_s = \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_s}-1}.$$ (8.43)

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: that is,

$$\sum_r \frac{1}{{\rm e}^{ \alpha+\beta \epsilon_r}-1} =N.$$ (8.44)

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

$$\ln Z = \alpha N - \sum_r\ln \left(1-{\rm e}^{-\alpha-\beta \epsilon_r}\right),$$ (8.45)

where use has been made of Equation (8.43).

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 (8.44) does not apply.