Classical limit of Quantum Statistical Distributions

According to Eqs. (145) and (147), the average number $n_j$ of noninteracting quantum particles in state $j$ is

$$\overline{n_j} = \frac{1}{ e^{\beta(\epsilon_j -\mu)} \mp 1},$$ (148)

where the minus sign corresponds to the Bose-Einstein distribution and the plus sign corresponds to the Fermi-Dirac distribution. In the limit when $\overline{n_j} \rightarrow 0$ the system is very "dilute", since there are much more energetically accessible states than particles. This limit is achieved for all states of energy $\epsilon_j$ when

$$e^{\beta(\epsilon_j -\mu)} >> 1.$$ (149)

Therefore, in the limit when $\overline{n_j} \rightarrow 0$,

$$\overline{n_j} \approx e^{-\beta(\epsilon_j -\mu)},$$ (150)

and the average number of particles is

$$\overline{N}= \sum_j \overline{n_j} \approx \sum_j e^{-\beta(\epsilon_j -\mu)},$$ (151)

so that

$$\beta \mu = \bar{N} - \sum_j e^{-\beta \epsilon_j}.$$ (152)

Moreover, according to Eqs. (150) and (151),

$$\frac{\overline{n_j}}{\overline{N}} = \frac{e^{-\beta \epsilon_j}}{\sum_j e^{-\beta \epsilon_j}},$$ (153)

which is the classical Boltzmann distribution. Therefore, in the limit when $\overline{n_j} \rightarrow 0$ both the Fermi-Dirac and the Bose-Einstein distributions converge to the classical Boltzmann distribution. Furthermore, according to Eqs. (143) and (146),

$$\Xi = \mp \sum_j (1 \mp e^{-\beta(\epsilon_j -\mu)}).$$ (154)

and in the limit when $e^{\beta(\epsilon_j -\mu)} >> 1$,

$$\Xi \approx \sum_j e^{-\beta(\epsilon_j -\mu)},$$ (155)

since

$$\lim_{x \rightarrow 0} (1+x) = x.$$ (156)

Therefore, according to Eqs. (151) and (155),

$$\Xi \approx \overline{N},$$ (157)

and according to Eqs. (157) and (85),

$$\overline{N} \approx ln Z + \beta \mu \overline{N}.$$ (158)

Substituting Eq. (152) into Eq. (157), we obtain

$$\overline{N} \approx ln Z + \overline{N} \overline{N} - \overline{N} \sum_j e^{-\beta \epsilon_j},$$ (159)

and according to the Stirling formula, introduced by Eq. (54),

$$Z = \frac{\left(\sum_j e^{-\beta \epsilon_j} \right)^{\overline{N}}}{\overline{N}!},$$ (160)

where the $1/N!$ factor, in Eq. (160), indicates that quantum particles remain indistinguishable even in the classical limit!