Editing Canonical ensemble (section)

==Properties==

{{unordered list
| ''Uniqueness'': The canonical ensemble is uniquely determined for a given physical system at a given temperature, and does not depend on arbitrary choices such as choice of coordinate system (classical mechanics), or basis (quantum mechanics), or of the zero of energy.<ref name="gibbs" /> The canonical ensemble is the only ensemble with constant {{mvar|N}}, {{mvar|V}}, and {{mvar|T}} that reproduces the [[fundamental thermodynamic relation]].<ref name="Gao2022">{{cite journal |last1= Gao |first1= Xiang |date= March 2022 |title= The Mathematics of the Ensemble Theory |journal= Results in Physics|volume= 34|pages= 105230|doi= 10.1016/j.rinp.2022.105230 |bibcode= 2022ResPh..3405230G |s2cid= 221978379 |doi-access= free |arxiv= 2006.00485 }}</ref>
| ''Statistical equilibrium'' (steady state): A canonical ensemble does not evolve over time, despite the fact that the underlying system is in constant motion. This is because the ensemble is only a function of a conserved quantity of the system (energy).<ref name="gibbs"/>
| ''Thermal equilibrium with other systems'': Two systems, each described by a canonical ensemble of equal temperature, brought into thermal contact<ref group=note>Thermal contact means that the systems are made able to exchange energy through an interaction. The interaction must be weak as to not significantly disturb the systems' microstates.{{clarify|date=January 2015}}</ref> will each retain the same ensemble and the resulting combined system is described by a canonical ensemble of the same temperature.<ref name="gibbs"/>
|''Maximum entropy'': For a given mechanical system (fixed {{math|''N''}}, {{math|''V''}}), the canonical ensemble average {{math|−⟨log ''P''⟩}} (the [[entropy]]) is the maximum possible of any ensemble with the same {{math|⟨''E''⟩}}.<ref name="gibbs"/>
| ''Minimum free energy'': For a given mechanical system (fixed {{math|''N''}}, {{math|''V''}}) and given value of {{math|''T''}}, the canonical ensemble average {{math|⟨''E'' + ''kT'' log ''P''⟩}} (the [[Helmholtz free energy]]) is the lowest possible of any ensemble.<ref name="gibbs"/> This is easily seen to be equivalent to maximizing the entropy.
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