Editing Phase transition (section)

===Ehrenfest classification===
[[Paul Ehrenfest]] classified phase transitions based on the behavior of the [[thermodynamic free energy]] as a function of other thermodynamic variables.<ref name="ReferenceA">{{cite journal|last1=Jaeger|first1=Gregg|title=The Ehrenfest Classification of Phase Transitions: Introduction and Evolution|journal=Archive for History of Exact Sciences|date=1 May 1998|volume=53|issue=1|pages=51–81|doi=10.1007/s004070050021|s2cid=121525126}}</ref> Under this scheme, phase transitions were labeled by the lowest derivative of the free energy that is discontinuous at the transition. ''First-order phase transitions'' exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable.<ref name = Blundell>{{Cite book | last = Blundell | first = Stephen J. |author2=Katherine M. Blundell | title = Concepts in Thermal Physics | publisher = Oxford University Press | year = 2008 | isbn = 978-0-19-856770-7}}</ref> The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the (inverse of the) first derivative of the free energy with respect to pressure. ''Second-order phase transitions'' are continuous in the first derivative (the [[#Order parameters|order parameter]], which is the first derivative of the free energy with respect to the external field, is continuous across the transition) but exhibit discontinuity in a second derivative of the free energy.<ref name = Blundell/> These include the ferromagnetic phase transition in materials such as iron, where the [[magnetization]], which is the first derivative of the free energy with respect to the applied magnetic field strength, increases continuously from zero as the temperature is lowered below the [[Curie temperature]]. The [[magnetic susceptibility]], the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classification scheme, there could in principle be third, fourth, and higher-order phase transitions. For example, the [[Gross–Witten–Wadia phase transition]] in 2-d lattice quantum chromodynamics is a third-order phase transition, and the [[Tracy–Widom distribution]] can be interpreted as a third-order transition.<ref>{{Citation |last1=Gross |first1=David J. |title=Possible third-order phase transition in the large N lattice gauge theory |journal=Physical Review D |date=1980 |volume=21 |issue=2 |pages=446–453 |doi=10.1103/PhysRevD.21.446|bibcode=1980PhRvD..21..446G }}</ref><ref>{{Cite journal |last1=Majumdar |first1=Satya N |last2=Schehr |first2=Grégory |date=2014-01-31 |title=Top eigenvalue of a random matrix: large deviations and third order phase transition |url=https://iopscience.iop.org/article/10.1088/1742-5468/2014/01/P01012 |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=2014 |issue=1 |pages=P01012 |doi=10.1088/1742-5468/2014/01/P01012 |issn=1742-5468|arxiv=1311.0580 |bibcode=2014JSMTE..01..012M |s2cid=119122520 }}</ref> The Curie points of many ferromagnetics is also a third-order transition, as shown by their specific heat having a sudden change in slope.<ref name=":0">{{Cite book |last=Pippard |first=Alfred B. |title=Elements of classical thermodynamics: for advanced students of physics |date=1981 |publisher=Univ. Pr |isbn=978-0-521-09101-5 |edition=Repr |location=Cambridge |pages=140–141}}</ref><ref>{{Cite journal |last=Austin |first=J. B. |date=November 1932 |title=Heat Capacity of Iron - A Review |journal=Industrial & Engineering Chemistry |volume=24 |issue=11 |pages=1225–1235 |doi=10.1021/ie50275a006 |issn=0019-7866}}</ref>

In practice, only the first- and second-order phase transitions are typically observed. The second-order phase transition was for a while controversial, as it seems to require two sheets of the Gibbs free energy to osculate exactly, which is so unlikely as to never occur in practice. [[Cornelis Jacobus Gorter|Cornelis Gorter]] replied the criticism by pointing out that the Gibbs free energy surface might have two sheets on one side, but only one sheet on the other side, creating a forked appearance.<ref>{{Cite journal |last=Jaeger |first=Gregg |date=1998-05-01 |title=The Ehrenfest Classification of Phase Transitions: Introduction and Evolution |url=https://doi.org/10.1007/s004070050021 |journal=Archive for History of Exact Sciences |language=en |volume=53 |issue=1 |pages=51–81 |doi=10.1007/s004070050021 |issn=1432-0657}}</ref> (<ref name=":0" /> pp. 146--150)

The Ehrenfest classification implicitly allows for continuous phase transformations, where the bonding character of a material changes, but there is no discontinuity in any free energy derivative. An example of this occurs at the [[supercritical liquid–gas boundaries]].

The first example of a phase transition which did not fit into the Ehrenfest classification was the exact solution of the [[Ising model]], discovered in 1944 by [[Lars Onsager]]. The exact specific heat differed from the earlier [[mean-field theory|mean-field]] approximations, which had predicted that it has a simple discontinuity at critical temperature. Instead, the exact specific heat had a logarithmic divergence at the critical temperature.<ref>{{cite book |last= Stanley|first= H. Eugene |authorlink=H. Eugene Stanley|date=1971 |title=Introduction to Phase Transitions and Critical Phenomena|location=Oxford |publisher=Clarendon Press}}</ref> In the following decades, the Ehrenfest classification was replaced by a simplified classification scheme that is able to incorporate such transitions.