|
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 explain 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:
- <math>
n_i = \frac {g_i} {\exp(\frac{\epsilon_i-\mu}{k T} ) - 1}
<math>
where:
- ni is the number of particles in state i
- gi is the degeneracy of state i
- εi 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 εi and containing a total of ni particles. Suppose each level contains gi 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 gi 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:
- <math>
w(n,2)=\frac{(n+1)!}{n!1!}
<math>
With a little thought it can be seen that the number of ways of distributing
n particles in 3 sublevels is w(n,3)=w(n,2)+w(n-1,2)+...+w(0,2) so that
- <math>
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!}
<math>
where we have used the following theorem involving binomial coefficients:
- <math>
\sum_{k=0}^n\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}
<math>
Continuing this process, we can see that w(n,g) is just a binomial coefficient
- <math>
w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}
<math>
The number of ways that a set of occupation numbers ni can be
realized is the product of the ways that each individual energy level can be populated:
- <math>
W = \prod_i w(n_i,g_i) = \prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!}
<math>
Following the same procedure used in deriving the Maxwell-Boltzmann distribution,
we wish to find the set of ni 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:
- <math>
f(n_i)=\ln(W)-\alpha(N-\sum n_i)+\beta(E-\sum n_i \epsilon_i)
<math>
Taking the derivative with respect to ni and setting the
result to zero and solving for ni yields the Bose-Einstein
population numbers. (Note: we have assumed that gi >>1)
- <math>
n_i = \frac{g_i}{e^{\alpha+\beta \epsilon_i}-1}
<math>
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:
- <math>\left.\right.
e^\alpha = e^{-\mu/kT} = 1/z
<math>
where μ is the chemical potential and z is the absolute activity.
See Also
| 
