Bose-Einstein and Fermi-Dirac Distributions

Consider a system consisting of $N$ quantum particles of a certain kind (e.g., bosons, or fermions with a certain spin). If the interaction of the particles is weak enough, each particle has its own motion which is independent of all others and system is an ideal gas of quantum particles. The quantum states allowed for this individual motion are the one-particle states $\vert j>$ that satisfy the eigenvalue problem

$$H \vert j > = \epsilon_j \vert j >,$$ (138)

where $\epsilon_j$ are the eigenvalues. Since identical particles are indistiguishable in quantum mechanics, each quantum state $\vert \xi >$ for the complete system is completely specified when the number of particles occupying each one-particle state is specified --i.e., the quantum numbers $\xi$ of the whole system are determined by the set of occupation numbers $n_1, n_2, n_3, ...$. The total energy of each quantum state $\vert \xi >$ is, therefore,

$$E_{\xi}=\sum_j n_j \epsilon_j.$$ (139)

Furthermore, since the quantum particles are indistinguishable,

$$\hat{P}_{jk} \vert\xi(1,2,3, ..., j, j+1, ..., k, k+1, ...)> = \pm \vert\xi(1,2,3, ..., k, j+1, ..., j, k+1, ...)>,$$ (140)

where $\hat{P}_{jk}$ is the operator that permutes particles $j$ and $k$. The plus sign, in Eq. (140), corresponds to a system of bosons (i.e., integer spin particles) and the minus sign corresponds to a system of fermions (i.e., half-integer spin particles). The Pauli Exclusion Principle is a consequence of the symmetry requirement introduced by Eq. (140). Such principle establishes that in a system of fermions with the same spin there cannot be two particles occupying the same spatial state and, therefore, $n_j=0, 1$. In a system of bosons, however, $n_j=0,1,2,...$ --i.e., there can be an arbitrary large number of particles in each state $j$. The grand canonical partition function for a system of indistigushable particles is defined, according to Eq. (81),

$$\Xi = \sum_{\xi} exp(-\beta E_{\xi} + \beta \mu n_{\xi}),$$ (141)

or in terms of occupation numbers $n_j$,

$$\Xi = \sum_{n_1, n_2, ...} exp(-\beta \sum_k \epsilon_k n_k + \beta \mu \sum_k n_k).$$ (142)

The gran canonical partition function for a system of fermions is

$$\Xi = \prod_k \sum_{n_k=0}^1 exp(-\beta \epsilon_k n_k + \beta \mu n_k)= \prod_k (1+e^{\beta(\mu-\epsilon_k)}),$$ (143)

due to the Pauli Exclusion Principle and Eq. (142). Therefore, the average occupation number

$$\overline{n_k} = \Xi^{-1} \sum_{n_1, n_2, ...} n_k e^{-\beta \sum_k \epsilon_k n_k + \beta \mu \sum_k n_k} = \partial \Xi/\partial(-\beta \epsilon_j),$$ (144)

is given by the following expression

$$\overline{n_k} = \frac{e^{\beta (\mu-\epsilon_k)}}{e^{\beta (\mu-\epsilon_k)}+1} = \frac{1}{1+e^{\beta (\epsilon_k-\mu)}},$$ (145)

which is the Fermi-Dirac distribution. Analogously, the gran canonical partition function for a system of bosons is

$$\Xi = \prod_k \sum_{n_k=0}^{\infty} e^{-\beta (\epsilon_k - \mu) n_k}= \prod_k \frac{1}{1-e^{\beta(\mu-\epsilon_k)}}.$$ (146)

Therefore, the average occupation number is given by the following expression

$$\overline{n_k} = \frac{1}{e^{\beta(\epsilon_k-\mu)}-1},$$ (147)

which is the Bose-Einstein distribution.  What is the Bose-Einstein condensation ?