Bose–Einstein statistics

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In statistical mechanics, Bose–Einstein statistics (or more colloquially B-E statistics) determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.

Fermi–Dirac and Bose–Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable". The quantum effects appear if the concentration of particles (N/V) ≥ nq (where nq is the quantum concentration). The quantum concentration is when the interparticle distance is equal to the thermal de Broglie wavelength i.e. when the wavefunctions of the particles are touching but not overlapping. As the quantum concentration depends on temperature; high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf. Fermi–Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), Bose–Einstein statistics apply to bosons. Both Fermi–Dirac and Bose–Einstein become Maxwell–Boltzmann statistics at high temperatures or low concentrations.

Maxwell–Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words the configuration of particle A in state 1 and particle B in state 2 is different from the case where particle B is in state 1 and particle A is in state 2. When this idea is carried out fully, it yields the proper (Boltzmann) distribution of particles in the energy states, but yields non-physical results for the entropy, as embodied in Gibbs paradox. These problems disappear when it is realized that all particles are in fact indistinguishable. Both of these distributions approach the Maxwell–Boltzmann distribution in the limit of high temperature and low density, without the need for any ad hoc assumptions. Maxwell–Boltzmann statistics are particularly useful for studying gases. Fermi–Dirac statistics are most often used for the study of electrons in solids. As such, they form the basis of semiconductor device theory and electronics.

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} {e^{(\epsilon_i-\mu)/kT} - 1}

with εi > μ and 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

This reduces to M-B statistics for energies ( εi-μ ) >> kT.

[edit] A 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. 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 ni  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 gi > > 1. Following the same procedure used in deriving the Maxwell–Boltzmann statistics, 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. The maxima of W and ln(W) occur at the value of Ni 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 gi > > 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 ni, and setting the result to zero and solving for ni 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}

Note that the above formula is sometimes written:

n_i = \frac{g_i}{e^{\epsilon_i/kT}/z-1}

where z = exp(μ / kT) is the absolute activity.

[edit] History

In the early 1920s Satyendra Nath Bose was intrigued by Einstein's theory of light waves being made of particles called photons. Bose was interested in deriving Planck's radiation formula, which Planck obtained largely by guessing. In 1900 Max Planck had derived his formula by manipulating the math to fit the empirical evidence. Using the particle picture of Einstein, Bose was able to derive the radiation formula by systematically developing a statistics of massless particles without the constraint of particle number conservation. Bose derived Planck's Law of Radiation by proposing different states for the photon. Instead of statistical independence of particles, Bose put particles into cells and described statistical independence of cells of phase space. Such systems allow two polarization states, and exhibit totally symmetric wavefunctions.

He was quite successful in that he developed a statistical law governing the behaviour pattern of photons. However he was not able to publish his work, because no journals in Europe would accept his paper being unable to understand it. Bose sent his paper to Einstein who saw the significance of it and he used his influence to get it published.

[edit] See also

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