AJAY KULKARNI: Bose-Einstein statistics

Tuesday, April 3, 2007

Bose-Einstein statistics

A Derivation of the Bose–Einstein distribution

Suppose we have a number of energy levels, labelled by index i, each level having energy ε_i and containing a total of n_i particles. Suppose each level contains g_i distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of g_i associated with level i is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel.

Let w(n,g) be the number of ways of distributing n particles among the g sublevels of an energy level. There is only one way of distributing n particles with one sublevel, therefore w(n,1) = 1. It's easy to see that there are n + 1 ways of distributing n particles in two sublevels which we will write as:

$w(n,2)=\frac{(n+1)!}{n!1!}.$

With a little thought it can be seen that the number of ways of distributing n particles in three sublevels is w(n,3) = w(n,2) + w(n−1,2) + ... + w(0,2) so that

$w(n,3)=\sum_{k=0}^n w(n-k,2) = \sum_{k=0}^n\frac{(n-k+1)!}{(n-k)!1!}=\frac{(n+2)!}{n!2!}$

where we have used the following theorem involving binomial coefficients:

$\sum_{k=0}^n\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.$

Continuing this process, we can see that w(n,g) is just a binomial coefficient

$w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}.$

The number of ways that a set of occupation numbers n_i can be realized is the product of the ways that each individual energy level can be populated:

$W = \prod_i w(n_i,g_i) = \prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!} \approx\prod_i \frac{(n_i+g_i)!}{n_i!(g_i)!}$

where the approximation assumes that $g i > > 1$ . Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of n_i for which $W$ is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of $W$ and $ln(W)$ occur at the value of $N i$ and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using Lagrange multipliers forming the function:

$f(n_i)=\ln(W)+\alpha(N-\sum n_i)+\beta(E-\sum n_i \epsilon_i)$

Using the $g i > > 1$ approximation and using Stirling's approximation for the factorials $\left(\ln(x!)\approx x\ln(x)-x\right)$ gives:

$f(n_i)=\sum_i (n_i + g_i) \ln(n_i + g_i) - n_i \ln(n_i) - g_i \ln (g_i) +\alpha(N-\sum n_i)+\beta(E-\sum n_i \epsilon_i)$

Taking the derivative with respect to n_i, and setting the result to zero and solving for n_i yields the Bose–Einstein population numbers:

$n_i = \frac{g_i}{e^{\alpha+\beta \epsilon_i}-1}$

It can be shown thermodynamically that β = 1/kT where k is Boltzmann's constant and T is the temperature, and that α = -μ/kT where μ is the chemical potential, so that finally:

$n_i = \frac{g_i}{e^{(\epsilon_i-\mu)/kT}-1}$