Editing Statistical mechanics (section)

===Three thermodynamic ensembles===
{{main|Ensemble (mathematical physics)|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}

There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.<ref name="gibbs"/> These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.
; [[Microcanonical ensemble]]
: describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.
; [[Canonical ensemble]]
: describes a system of fixed composition that is in [[thermal equilibrium]] with a [[heat bath]] of a precise [[thermodynamic temperature|temperature]]. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.
; [[Grand canonical ensemble]]
: describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise [[chemical potential]]s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.

For systems containing many particles (the [[thermodynamic limit]]), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used.<ref name="Reif" />{{rp|227}} The Gibbs theorem about equivalence of ensembles<ref>{{cite journal |doi=10.1007/s10955-015-1212-2|title=Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels|journal=Journal of Statistical Physics|volume=159|issue=5|pages=987–1016|year=2015|last1=Touchette|first1=Hugo|arxiv=1403.6608|bibcode=2015JSP...159..987T|s2cid=118534661}}</ref> was developed into the theory of [[concentration of measure]] phenomenon,<ref>{{cite book |doi=10.1090/surv/089 |title=The Concentration of Measure Phenomenon |series=Mathematical Surveys and Monographs |date=2005 |volume=89 |isbn=978-0-8218-3792-4 |url=http://www.gbv.de/dms/bowker/toc/9780821837924.pdf }}{{pn|date=April 2024}}</ref> which has applications in many areas of science, from functional analysis to methods of [[artificial intelligence]] and [[big data]] technology.<ref>{{cite journal |last1=Gorban |first1=A. N. |last2=Tyukin |first2=I. Y. |title=Blessing of dimensionality: mathematical foundations of the statistical physics of data |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 April 2018 |volume=376 |issue=2118 |pages=20170237 |doi=10.1098/rsta.2017.0237 |pmid=29555807 |pmc=5869543 |arxiv=1801.03421 |bibcode=2018RSPTA.37670237G }}</ref>

Important cases where the thermodynamic ensembles ''do not'' give identical results include:
* Microscopic systems.
* Large systems at a phase transition.
* Large systems with long-range interactions.
In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.<ref name="tolman" />

{| class="wikitable" style="text-align: center"
|+ Thermodynamic ensembles<ref name="gibbs" />
|-
!
! [[Microcanonical ensemble|Microcanonical]]
! [[Canonical ensemble|Canonical]]
! [[Grand canonical ensemble|Grand canonical]]
|-
! Fixed variables
| <math>E, N, V</math>
| <math>T, N, V</math>
| <math>T, \mu, V</math>
|-
! rowspan="2" | Microscopic features
| Number of [[Microstate (statistical mechanics)|microstates]]
| [[Canonical partition function]]
| [[Grand partition function]]
|-
| <math>W</math>
| <math>Z = \sum_k e^{- E_k / k_B T}</math>
| <math>\mathcal Z = \sum_k e^{ -(E_k - \mu N_k) /k_B T}</math>
|-
! rowspan="2" | Macroscopic function
| [[Boltzmann entropy]]
| [[Helmholtz free energy]]
| [[Grand potential]]
|-
| <math>S = k_B \log W</math>
| <math>F = - k_B T \log Z</math>
| <math>\Omega =- k_B T \log \mathcal Z </math>
|}