Particle statistics

Particle statistics
Maxwell-Boltzmann statistics
Bose-Einstein statistics
Fermi-Dirac statistics
Parastatistics
Anyonic statistics
Braid statistics
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Particle statistics refers to the particular description of particles in statistical mechanics. The three main types of particle statistics are:

$\bar{n} = \frac{1}{e^{\left(\epsilon-\mu\right)/k T}}$

$\bar{n} = \frac{1}{e^{\left(\epsilon-\mu\right)/k T}+1}$

$\bar{n} = \frac{1}{e^{\left(\epsilon-\mu\right)/k T}-1}$

The difference between these three kinds of statistics is due to the following facts:

In classical physics, particles are treated as different entities, which can be distinguished from each other.
In quantum mechanics, bosons can be distinguished from each other only due to their different physical state, and bosons in the same physical state cannot be distinguished from each other. Thus, the situation in which photon A is in physical state 1, and photon B is in physical state 2, is the same as the situation in which photon A is in physical state 2, and photon B is in physical state 1. While in classical mechanics these will be counted as two different situations, in quantum mechanics they will be counted as one. As a consequence of this, bosons act as if they prefer to be in the same physical state.
In quantum mechanics, two fermions cannot be in the same physical state.

Mathematically, this is a result of describing bosons by commuting operators, and fermions by anticommuting operators.

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