Derivation of Bose Distribution

This document will try to document the way we compute the probability of occupation by n indistinguishable particles of the ith energy level which has g sub-levels. I have found that the best way to keep track of the occupations of the various g sub-level populations is to assign a number to each sub level and then tabulate the sums of these numbers for all possible indistinguishable sets of sums. This can be motivated by assuming that each of the sub-levels has a very small magnetic energy splitting and then the sum of these energy values will distinguish it from other sub-levels. In the end, we can simply remove the splitting but the sub-level distribution will remain.

The number of different sums we can get for n particles assigned to levels with g degeneracy is the binomial factor

$w (n, g) = \frac{(n + g - 1)!}{n! (g - 1)!} \equiv (\begin{array}{l} n + g - 1 \\ g - 1 \end{array})$ (1.1)

If the Bose character of the particle is to be interesting, then we should have many more particles than there are degeneracies, i.e. n>>g. The opposite situation is when n<<g very low density gas which characterizes Maxwell-Boltzmann statistics.

The total number of ways that all N indistinguishable particles with n_i in each level with g_i sub-levels or degeneracies is the product of all the binomial factors for the n_i+g_i-1 terms

$W = \prod_{i} w (n_{i}, g_{i}) = \prod_{i} \frac{(n_{i} + g_{i} - 1)!}{n_{i}! (g_{i} - 1)!}$

(1.2)

Use Stirling's approximation for the factorials:

$x! = x^{x} e^{- x} \sqrt{2 π x}$

(1.3)

$\frac{(n_{i} + g_{i} - 1)!}{n_{i}! (g_{i} - 1)!} = \frac{{(n_{i} + g_{i} - 1)}^{(n_{i} + g_{i} - 1)} e^{- (n_{i} + g_{i} - 1)} \sqrt{2 π (n_{i} + g_{i} - 1)}}{n^{n_{i}} e^{- n_{i}} \sqrt{2 π n_{i}} (g_{i} - 1)!}$

(1.4)

The number of particles, n_i, in each energy level ε_i will be such that the value of W is maximized taking into account that the total number of particles is constant at N and the total energy is constant at E.

We wish to find the values of each n_i for which W is a maximum taking into account that the total number of particles, N, is constant and the total energy is constant. To find n_i while keeping N and E constant, we use Lagrange multipliers for the particle sums and the total energy. Then the equation from which to find n_i becomes:

$\frac{\partial}{\partial n_{i}} [w (n_{i}, g_{i}) + α (N - \sum_{i} n_{i}) + β (E - \sum_{i} n_{i} ε_{i})] = 0$

(1.5)

It is easier to find the maximum if we first take the natural logarithm of the expression for w.

Using equation (1.4) in equation (1.5) we obtain:

$\begin{array}{l} \ln [\frac{(n_{i} + g_{i} - 1)!}{n_{i}! (g_{i} - 1)!}] \approx \ln [\frac{{(n_{i} + g_{i} - 1)}^{(n_{i} + g_{i} - 1)} e^{- (n_{i} + g_{i} - 1)} \sqrt{2 π (n_{i} + g_{i} - 1)}}{n^{n_{i}} e^{- n_{i}} \sqrt{2 π n_{i}} (g_{i} - 1)!}] \\ \approx (n_{i} + g_{i} - 1) l n (n_{i} + g_{i} - 1) - (n_{i} + g_{i} - 1) + n_{i} \ln (n_{i}) - n_{i} - \ln [(g_{i} - 1)!] \end{array}$

(1.6)

where I've dropped the square root terms. If we also make the assumption that n_i>>g_i then we can further simplify the above expression. Let’s rename the factorial expression on the left to be w(n_i,g_i)

$\ln [w_{i} (n_{i}, g_{i})] \approx (n_{i} + g_{i} - 1) l n (n_{i} + g_{i} - 1) - n_{i} \ln (n_{i}) - l n [(g_{i} - 1)!]$

(1.7)

We will need the derivative of w_i with respect to n_i:

$\frac{\partial [\ln (w)]}{\partial n_{i}} = l n (n_{i} + g_{i} - 1) + 1 - \ln (n_{i}) + 1 = l n (\frac{n_{i} + g_{i} - 1}{n_{i}})$

(1.8)

Including the particle sums and energy sum derivative terms in equation (1.5) we have:

$l n (\frac{n_{i} + g_{i} - 1}{n_{i}}) - α - β ε_{i} = 0$

(1.9)

taking the exponential of both sides we have:

$(\frac{n_{i} + g_{i} - 1}{n_{i}}) = e^{(α + β ε_{i})}$

(1.10)

Solving for n_iwe obtain:

$\begin{array}{l} n_{i} + g_{i} - 1 = n_{i} e^{(α + β ε_{i})} \\ n_{i} [1 - e^{(α + β ε_{i})}] = - (g_{i} - 1) \\ n_{i} = \frac{g_{i} - 1}{e^{(α + β ε_{i})} - 1} \end{array}$

(1.11)

Equation (1.11) has the form of the Bose distribution and is better recognized with the substitution

$\begin{array}{l} c h e m i c a l p o t e n t i a l μ : α = - \frac{μ}{k_{B} T} \\ B o l t z m a n n F a c t o r : β = \frac{1}{k_{B} T} \end{array}$

(1.12)

so that we have:

$n_{i} = \frac{(g_{i} - 1)}{e^{(\frac{ε_{i} - μ}{k_{B} T})} - 1}$

(1.13)