Editing Microcanonical ensemble (section)

=== Phase transitions ===
Under their strict definition, [[phase transition]]s correspond to [[analytic function|nonanalytic]] behavior in the thermodynamic potential or its derivatives.<ref name="Goldenfeld">[[Nigel Goldenfeld]]; ''Lectures on Phase Transitions and the Renormalization Group'', Frontiers in Physics 85, Westview Press (June, 1992) {{ISBN|0-201-55409-7}}</ref> Using this definition, phase transitions in the microcanonical ensemble can occur in systems of any size. This contrasts with the canonical and grand canonical ensembles, for which phase transitions can occur only in the [[thermodynamic limit]] – i.e., in systems with infinitely many degrees of freedom.<ref name="Goldenfeld"/><ref name="DunkelHilbert2006">{{cite journal|last1=Dunkel|first1=Jörn|last2=Hilbert|first2=Stefan|title=Phase transitions in small systems: Microcanonical vs. canonical ensembles|journal=Physica A: Statistical Mechanics and Its Applications|volume=370|issue=2|year=2006|pages=390–406|issn=0378-4371|doi=10.1016/j.physa.2006.05.018|arxiv=cond-mat/0511501|bibcode=2006PhyA..370..390D |s2cid=13900006 }}</ref> Roughly speaking, the reservoirs defining the canonical or grand canonical ensembles introduce fluctuations that "smooth out" any nonanalytic behavior in the free energy of finite systems. This smoothing effect is usually negligible in macroscopic systems, which are sufficiently large that the free energy can approximate nonanalytic behavior exceedingly well. However, the technical difference in ensembles may be important in the theoretical analysis of small systems.<ref name="DunkelHilbert2006"/>