Bose-Einstein statistics

Bose-Einstein statistics are used to describe a gas of indistinguishable bosons. Particles with integral spin must necessarily have wave functions which are symmetric under particle exchange. That is, two configurations which differ by only exchanging the positions and velocities of a pair of bosons are given identical weights.

The average number of particles in state s is given by

<ns> = 1/(exp((Es-u)/ kBT) - 1)

where Es is the energy of a particle in s, u is the chemical potential, T is the temperature, and kB is Boltzmann's constant. Compare this to Fermi-Dirac statistics where the -1 is replaced by +1. In the limit where exp((Es-u)/ kBT) >> 1, either due to large T or large u, then the quantum nature of the gas is unimportant and the system is described by classical Maxwell-Boltzmann statistics.

On the other hand, for very cold temperatures (and in a finite volume), the number of particles allowed to be in excited energy states is bounded. Therefore, if additional particles are put into the system, they are pushed into the ground state. Under these circumstances it is possible that a macroscopically large number of particles can all be in a single quantum state (with E0 = 0). This phenomenon is called Bose-Einstein condensation.

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