Editing Canonical ensemble (section)

== Free energy, ensemble averages, and exact differentials ==

* The partial derivatives of the function {{math|''F''(''N'', ''V'', ''T'')}} give important canonical ensemble average quantities:
** the average pressure is<ref name="gibbs"/> <math display="block"> \langle p \rangle = -\frac{\partial F} {\partial V}, </math>
** the [[Gibbs entropy]] is<ref name="gibbs"/> <math display="block"> S = -k \langle \log P \rangle = - \frac{\partial F} {\partial T}, </math>
** the partial derivative {{math|∂''F''/∂''N''}} is approximately related to [[chemical potential]], although the concept of chemical equilibrium does not exactly apply to canonical ensembles of small systems.<ref group=note>Since {{math|''N''}} is an integer, this "derivative" actually refers to a [[finite difference]] expression such as {{math|''F''(''N'') − ''F''(''N'' − 1)}}, or {{math|''F''(''N'' + 1) − ''F''(''N'')}}, or {{math|[''F''(''N'' + 1) − ''F''(''N'' − 1)]/2}}. These finite difference expressions are equivalent only in the thermodynamic limit (very large {{math|''N''}}).</ref>
** and the average energy is<ref name="gibbs"/> <math display="block"> \langle E \rangle = F + ST.</math>
* ''Exact differential'': From the above expressions, it can be seen that the function {{math|''F''(''V'', ''T'')}}, for a given {{math|''N''}}, has the [[exact differential]]<ref name="gibbs"/> <math display="block"> dF = - S \, dT - \langle p\rangle \, dV .</math>
* ''First law of thermodynamics'': Substituting the above relationship for {{math|⟨''E''⟩}} into the exact differential of {{math|''F''}}, an equation similar to the [[first law of thermodynamics]] is found, except with average signs on some of the quantities:<ref name="gibbs"/> <math display="block"> d\langle E \rangle = T \, dS - \langle p\rangle \, dV .</math>
* ''[[Thermal fluctuations|Energy fluctuations]]'': The energy in the system has uncertainty in the canonical ensemble. The [[variance]] of the energy is<ref name="gibbs"/> <math display="block"> \langle E^2 \rangle - \langle E \rangle^2 = k T^2 \frac{\partial \langle E \rangle}{\partial T}.</math>