Editing Phase transition (section)

==Classifications==

===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|url-access=subscription }}</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.

===Modern classifications===
In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes:<ref name="ReferenceA"/>

'''First-order phase transitions''' are those that involve a [[latent heat]]. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume. During this process, the temperature of the system will stay constant as heat is added: the system is in a "mixed-phase regime" in which some parts of the system have completed the transition and others have not.<ref>Faghri, A., and Zhang, Y., [https://books.google.com/books?id=bxndY2KSuQsC&q=Transport+Phenomena+in+Multiphase+Systems ''Transport Phenomena in Multiphase Systems''], Elsevier, Burlington, MA, 2006,</ref><ref>Faghri, A., and Zhang, Y., [https://www.springer.com/gp/book/9783030221362 ''Fundamentals of Multiphase Heat Transfer and Flow''], Springer, New York, NY, 2020</ref> 

Familiar examples are the melting of ice or the boiling of water (the water does not instantly turn into [[water vapor|vapor]], but forms a [[turbulence|turbulent]] mixture of liquid water and vapor bubbles). [[Yoseph Imry]] and Michael Wortis showed that [[quenched disorder]] can broaden a first-order transition. That is, the transformation is completed over a finite range of temperatures, but phenomena like supercooling and superheating survive and hysteresis is observed on thermal cycling.<ref>{{cite journal | last1 = Imry | first1 = Y. | last2 = Wortis | first2 = M. | year = 1979 | title =  Influence of quenched impurities on first-order phase transitions| journal = Phys. Rev. B | volume = 19 | issue = 7| pages = 3580–3585 | doi=10.1103/physrevb.19.3580|bibcode = 1979PhRvB..19.3580I }}</ref><ref name="KumarPramanik2006">{{cite journal|last1=Kumar|first1=Kranti|last2=Pramanik|first2=A. K.|last3=Banerjee|first3=A.|last4=Chaddah|first4=P.|last5=Roy|first5=S. B.|last6=Park|first6=S.|last7=Zhang|first7=C. L.|last8=Cheong|first8=S.-W.|title=Relating supercooling and glass-like arrest of kinetics for phase separated systems: DopedCeFe2and(La,Pr,Ca)MnO3|journal=Physical Review B|volume=73|issue=18|page=184435|year=2006|issn=1098-0121|doi=10.1103/PhysRevB.73.184435|arxiv = cond-mat/0602627 |bibcode = 2006PhRvB..73r4435K |s2cid=117080049}}</ref><ref name="PasquiniDaroca2008">{{cite journal|last1=Pasquini|first1=G.|last2=Daroca|first2=D. Pérez|last3=Chiliotte|first3=C.|last4=Lozano|first4=G. S.|last5=Bekeris|first5=V.|title=Ordered, Disordered, and Coexistent Stable Vortex Lattices inNbSe2Single Crystals|journal=Physical Review Letters|volume=100|issue=24|page=247003|year=2008|issn=0031-9007|doi=10.1103/PhysRevLett.100.247003|pmid=18643617|bibcode=2008PhRvL.100x7003P|arxiv=0803.0307|s2cid=1568288}}</ref>

'''{{va|Second-order phase transition}}s''' are also called ''"continuous phase transitions"''. They are characterized by a divergent susceptibility, an infinite [[Correlation function (statistical mechanics)|correlation length]], and a [[power law]] decay of correlations near [[Critical point (thermodynamics)|criticality]]. Examples of second-order phase transitions are the [[Ferromagnetism|ferromagnetic]] transition, superconducting transition (for a [[Type-I superconductor]] the phase transition is second-order at zero external field and for a [[Type-II superconductor]] the phase transition is second-order for both normal-state–mixed-state and mixed-state–superconducting-state transitions) and the [[superfluid]] transition. In contrast to viscosity, thermal expansion and heat capacity of amorphous materials show a relatively sudden change at the glass transition temperature<ref name="J. Non-Cryst 2013">{{cite journal | last1 = Ojovan | first1 = M.I. | year = 2013 | title = Ordering and structural changes at the glass-liquid transition | journal = J. Non-Cryst. Solids | volume = 382 | pages = 79–86 | doi = 10.1016/j.jnoncrysol.2013.10.016 |bibcode = 2013JNCS..382...79O }}</ref> which enables accurate detection using [[differential scanning calorimetry]] measurements.  [[Lev Landau]] gave a [[Phenomenology (particle physics)|phenomenological]] [[Landau theory|theory]] of second-order phase transitions.

Apart from isolated, simple phase transitions, there exist transition lines as well as [[multicritical point]]s, when varying external parameters like the magnetic field or composition.

Several transitions are known as ''infinite-order phase transitions''.
They are continuous but break no [[#Symmetry|symmetries]]. The most famous example is the [[Kosterlitz–Thouless transition]] in the two-dimensional [[XY model]]. Many [[quantum phase transition]]s, e.g., in [[two-dimensional electron gas]]es, belong to this class.

The [[glass transition|liquid–glass transition]] is observed in many [[polymers]] and other liquids that can be [[supercooling|supercooled]] far below the melting point of the crystalline phase. This is atypical in several respects. It is not a transition between thermodynamic ground states: it is widely believed that the true ground state is always crystalline. Glass is a ''[[quenched disorder]]'' state, and its entropy, density, and so on, depend on the thermal history. Therefore, the glass transition is primarily a dynamic phenomenon: on cooling a liquid, internal degrees of freedom successively fall out of equilibrium. Some theoretical methods predict an underlying phase transition in the hypothetical limit of infinitely long relaxation times.<ref>Gotze, Wolfgang. "Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory."</ref><ref>{{cite journal | last1 = Lubchenko | first1 = V. Wolynes | last2 = Wolynes | first2 = Peter G. | year = 2007 | title = Theory of Structural Glasses and Supercooled Liquids | journal = Annual Review of Physical Chemistry | volume = 58 | pages = 235–266 | doi=10.1146/annurev.physchem.58.032806.104653| pmid = 17067282 |arxiv = cond-mat/0607349 |bibcode = 2007ARPC...58..235L | s2cid = 46089564 }}</ref> No direct experimental evidence supports the existence of these transitions.