Editing Ensemble (mathematical physics) (section)

== Main types ==
[[File:Statistical Ensembles.png|600px|thumb|right|Visual representation of five statistical ensembles (from left to right): [[microcanonical ensemble]], [[canonical ensemble]], [[grand canonical ensemble]], [[isobaric-isothermal ensemble]], [[isoenthalpic-isobaric ensemble]] ]]
The study of thermodynamics is concerned with systems that appear to human perception to be "static" (despite the motion of their internal parts), and which can be described simply by a set of macroscopically observable variables. These systems can be described by statistical ensembles that depend on a few observable parameters, and which are in statistical equilibrium. Gibbs noted that different macroscopic constraints lead to different types of ensembles, with particular statistical characteristics.
<blockquote>''"We may imagine a great number of systems of the same nature, but differing in the configurations and velocities which they have at a given instant, and differing in not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities..." J. W. Gibbs'' (1903)<ref>{{Cite book |last=Gibbs |first=J.W. |title=The Collected Works, Vol. 2. |publisher=Longmans. |year=1928 |location=Green & Co, London, New York}}</ref></blockquote>
Three important thermodynamic ensembles were defined by Gibbs:<ref name="gibbs" />
* ''[[Microcanonical ensemble]]'' (or ''NVE ensemble'') —a statistical ensemble where the total energy of the system and the number of particles in the system are each fixed to particular values; each of the members of the ensemble are required to have the same total energy and particle number. The system must remain totally isolated (unable to exchange energy or particles with its environment) in order to stay in statistical equilibrium.<ref name="gibbs"/>
* ''[[Canonical ensemble]]'' (or ''NVT ensemble'')—a statistical ensemble where the energy is not known exactly but the number of particles is fixed. In place of the energy, the [[temperature]] is specified. The canonical ensemble is appropriate for describing a closed system which is in, or has been in, weak [[thermal contact]] with a heat bath. In order to be in statistical equilibrium, the system must remain totally closed (unable to exchange particles with its environment) and may come into weak thermal contact with other systems that are described by ensembles with the same temperature.<ref name="gibbs"/>
* ''[[Grand canonical ensemble]]'' (or ''μVT ensemble'')—a statistical ensemble where neither the energy nor particle number are fixed. In their place, the temperature and [[chemical potential]] are specified. The grand canonical ensemble is appropriate for describing an open system: one which is in, or has been in, weak contact with a reservoir (thermal contact, chemical contact, radiative contact, electrical contact, etc.). The ensemble remains in statistical equilibrium if the system comes into weak contact with other systems that are described by ensembles with the same temperature and chemical potential.<ref name="gibbs"/>

The calculations that can be made using each of these ensembles are explored further in their respective articles.
Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived.
For example, in the reaction ensemble, particle number fluctuations are only allowed to occur according to the [[stoichiometry]] of the [[chemical reaction]]s which are present in the system.<ref>{{cite journal | last1=Heath Turner | first1=C. | last2=Brennan | first2=John K. | last3=Lísal | first3=Martin | last4=Smith | first4=William R. | last5=Karl Johnson | first5=J. | last6=Gubbins | first6=Keith E. | title=Simulation of chemical reaction equilibria by the reaction ensemble Monte Carlo method: a review | journal=Molecular Simulation | publisher=Informa UK Limited | volume=34 | issue=2 | year=2008 | issn=0892-7022 | doi=10.1080/08927020801986564 | pages=119–146}}</ref> 

=== Equivalence ===

In thermodynamic limit all ensembles should produce identical observables due to [[Legendre_transformation| Legendre transforms]], deviations to this rule occurs under conditions that state-variables are non-convex, such as small molecular measurements. <ref name="pre_eq">{{cite journal|url= https://journals.aps.org/pre/abstract/10.1103/PhysRevE.79.051118 |title= "Ensemble inequivalence in single-molecule experiments"|journal= Physical Review E|date= 18 May 2009|volume= 79|issue= 5|page= 051118 |doi=10.1103/PhysRevE.79.051118 |language=en-US|access-date=2024-03-03|last1= Süzen|first1= M |last2= Sega |first2= M|last3= Holm|first3= C|arxiv= 0810.3407|bibcode= 2009PhRvE..79e1118S}}</ref>