Editing Boltzmann distribution (section)

== In statistical mechanics ==
{{main|Canonical ensemble|Maxwell–Boltzmann statistics}}

The Boltzmann distribution appears in [[statistical mechanics]] when considering closed systems of fixed composition that are in [[thermal equilibrium]] (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble. Some special cases (derivable from the canonical ensemble) show the Boltzmann distribution in different aspects:

; [[Canonical ensemble]] (general case)
: The [[canonical ensemble]] gives the [[probabilities]] of the various possible states of a closed system of fixed volume, in thermal equilibrium with a [[heat bath]]. The canonical ensemble has a state probability distribution with the Boltzmann form.
; Statistical frequencies of subsystems' states (in a non-interacting collection)
: When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the [[statistical frequency]] of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the [[expectation value|expected]] statistical frequency distribution of subsystem states has the Boltzmann form.
; [[Maxwell–Boltzmann statistics]] of classical gases (systems of non-interacting particles)
: In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. [[Maxwell–Boltzmann statistics]] give the expected number of particles found in a given single-particle state, in a [[classical mechanics|classical]] gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.

Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:
* When a system is in thermodynamic equilibrium with respect to both energy exchange ''and particle exchange'', the requirement of fixed composition is relaxed and a [[grand canonical ensemble]] is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a [[microcanonical ensemble]] applies instead.
* If the subsystems within a collection ''do'' interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an [[analytical solution]].<ref>A classic example of this is [[magnetic ordering]]. Systems of non-interacting [[Spin (physics)|spins]] show [[paramagnetic]] behaviour that can be understood with a single-particle canonical ensemble (resulting in the [[Brillouin function]]). Systems of ''interacting'' spins can show much more complex behaviour such as [[ferromagnetism]] or [[antiferromagnetism]].</ref> The canonical ensemble can however still be applied to the ''collective'' states of the entire system considered as a whole, provided the entire system is in thermal equilibrium.
* With ''[[quantum mechanics|quantum]]'' gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by [[Fermi–Dirac statistics]] or [[Bose–Einstein statistics]], depending on whether the particles are [[fermion]]s or [[boson]]s, respectively.