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Categories: Statistical mechanics | Quantum mechanics | Quantum field theory | Probability distributions

Bose-Einstein statistics

For other topics related to Einstein see Einstein (disambig)

In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.

Bose-Einstein (or B-E) statistics are closely related to Maxwell-Boltzmann statistics (M-B) and Fermi-Dirac statistics (F-D). While F-D statistics holds for fermions, M-B statistics holds for classical particles, i.e. identical but distinguishable particles, and represents the classical or high-temperature limit of both F-D and B-E statistics. (M-B, B-E, and F-D statistics are all derived from the Boltzmann factor probability weight applied to the problem of classical particles and discrete energy quanta with boson/fermion behavior, respectively.)

Bosons, unlike fermions, are not subject to the Pauli exclusion principle: an unlimited number of particles may occupy the same state at the same time. This explains why, at low temperatures, bosons can behave very differently than fermions; all the particles will tend to congregate together at the same lowest-energy state, forming what is known as a Bose-Einstein condensate.

B-E statistics was introduced for photons in 1920 by Bose and generalized to atoms by Einstein in 1924.

The expected number of particles in an energy state i for B-E statistics is:

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

where:

n_i is the number of particles in state i

g_i is the degeneracy of state i

ε_i is the energy of the i-th state

μ is the chemical potential

k is Boltzmann's constant

T is absolute temperature

exp is the exponential function

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. Its 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:

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

Following the same procedure used in deriving the Maxwell-Boltzmann distribution, 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. We constrain our solution using Lagrange multipliers forming the function:

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

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. (Note: we have assumed that g_i >>1)

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

It can be shown thermodynamically that β=1/kT where k is Boltzmann's constant and T is the temperature. The term containing α is variously written:

$\left.\right. e^\alpha = e^{-\mu/kT} = 1/z$

where μ is the chemical potential and z is the absolute activity.

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