Editing Maxwell–Boltzmann statistics (section)

==Applicability==
[[File:Quantum and classical statistics.png|500px|thumb|right|Equilibrium thermal distributions for particles with integer spin (bosons), half integer spin (fermions), and classical (spinless) particles. Average occupancy <math>\langle n\rangle</math> is shown versus energy <math>\epsilon</math> relative to the system chemical potential <math>\mu</math>, where <math>T</math> is the system temperature, and <math>k_B</math> is the Boltzmann constant.]]

Maxwell–Boltzmann statistics is used to derive the [[Maxwell–Boltzmann distribution]] of an ideal gas. However, it can also be used to extend that distribution to particles with a different [[energy–momentum relation]], such as relativistic particles (resulting in [[Maxwell–Jüttner distribution]]), and to other than three-dimensional spaces.

Maxwell–Boltzmann statistics is 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 in which particle ''B'' is in state 1 and particle ''A'' is in state 2. This assumption leads to the proper (Boltzmann) statistics of particles in the energy states, but yields non-physical results for the entropy, as embodied in the [[Gibbs paradox]].

At the same time, there are no real particles that have the characteristics required by Maxwell–Boltzmann statistics. Indeed, the Gibbs paradox is resolved if we treat all particles of a certain type (e.g., electrons, protons, photons, etc.) as principally indistinguishable. Once this assumption is made, the particle statistics change. The change in entropy in the [[entropy of mixing]] example may be viewed as an example of a non-extensive entropy resulting from the distinguishability of the two types of particles being mixed.

Quantum particles are either bosons (following [[Bose–Einstein statistics]]) or fermions (subject to the [[Pauli exclusion principle]], following instead [[Fermi–Dirac statistics]]). Both of these quantum statistics approach the Maxwell–Boltzmann statistics in the limit of high temperature and low particle density.