Editing Phase transition (section)

==Characteristic properties==

===Phase coexistence===
A disorder-broadened  first-order transition occurs over a finite range of temperatures where the fraction of the low-temperature equilibrium phase grows from zero to one (100%) as the temperature is lowered. This continuous variation of the coexisting fractions with temperature raised interesting possibilities. On cooling, some liquids vitrify into a glass rather than transform to the equilibrium crystal phase. This happens if the cooling rate is faster than a critical cooling rate, and is attributed to the molecular motions becoming so slow that the molecules cannot rearrange into the crystal positions.<ref>{{cite journal | year = 1995 | title =  Metallic Glasses| journal = Science | volume = 267 | issue = 5206| pages = 1947–1953 |bibcode = 1995Sci...267.1947G |doi = 10.1126/science.267.5206.1947 | pmid =  17770105| last1 =  Greer| first1 =  A. L.| s2cid = 220105648}}</ref> This slowing down happens below a glass-formation temperature ''T''<sub>g</sub>, which may depend on the applied pressure.<ref name="J. Non-Cryst 2013"/><ref>{{cite journal | last1 = Tarjus | first1 = G. | year = 2007 | title =  Materials science: Metal turned to glass| journal = Nature | volume = 448 | issue = 7155| pages = 758–759 | doi=10.1038/448758a| pmid = 17700684 |bibcode = 2007Natur.448..758T | s2cid = 4410586 | doi-access = free }}</ref> If the first-order freezing transition occurs over a range of temperatures, and ''T''<sub>g</sub> falls within this range, then there is an interesting possibility that the transition is arrested when it is partial and incomplete. Extending these ideas to first-order magnetic transitions being arrested at low temperatures, resulted in the observation of incomplete magnetic transitions, with two magnetic phases coexisting, down to the lowest temperature. First reported in the case of a ferromagnetic to anti-ferromagnetic transition,<ref name="ManekarChaudhary2001">{{cite journal |last1=Manekar |first1=M. A. |last2=Chaudhary |first2=S. |last3=Chattopadhyay |first3=M. K. |last4=Singh |first4=K. J. |last5=Roy |first5=S. B. |last6=Chaddah |first6=P. |title=First-order transition from antiferromagnetism to ferromagnetism inCe(Fe<sub>0.96</sub>Al<sub>0.04</sub>)<sub>2</sub> |journal=Physical Review B |volume=64 |issue=10 |page=104416 |year=2001 |issn=0163-1829 |doi=10.1103/PhysRevB.64.104416 |arxiv=cond-mat/0012472 |bibcode=2001PhRvB..64j4416M|s2cid=16851501 }}</ref> such persistent phase coexistence has now been reported across a variety of first-order magnetic transitions. These include colossal-magnetoresistance manganite materials,<ref>{{cite journal|doi=10.1088/0953-8984/18/49/L02|arxiv = cond-mat/0611152 |bibcode = 2006JPCM...18L.605B |title = Coexisting tunable fractions of glassy and equilibrium long-range-order phases in manganites |journal = Journal of Physics: Condensed Matter |volume = 18 |issue = 49 |pages = L605 |year = 2006 |last1 = Banerjee |first1 = A. |last2 = Pramanik |first2 = A. K. |last3 = Kumar |first3 = Kranti |last4 = Chaddah |first4 = P. |s2cid = 98145553 }}</ref><ref>{{cite journal |author=Wu W. |author2=Israel C. |author3=Hur N. |author4=Park S. |author5=Cheong S. W. |author6=de Lozanne A. | year = 2006 | title =  Magnetic imaging of a supercooling glass transition in a weakly disordered ferromagnet| journal = Nature Materials | volume = 5 | issue = 11| pages = 881–886 |bibcode = 2006NatMa...5..881W |doi = 10.1038/nmat1743 | pmid = 17028576 | s2cid = 9036412 }}</ref> magnetocaloric materials,<ref name="RoyChattopadhyay2006">{{cite journal |last1=Roy |first1=S. B. |last2=Chattopadhyay |first2=M. K. |last3=Chaddah |first3=P. |last4=Moore |first4=J. D. |last5=Perkins |first5=G. K. |last6=Cohen |first6=L. F. |last7=Gschneidner |first7=K. A. |last8=Pecharsky |first8=V. K. |title=Evidence of a magnetic glass state in the magnetocaloric material Gd<sub>5</sub>Ge<sub>4</sub> |journal=Physical Review B |volume=74 |issue=1 |page=012403 |year=2006 |issn=1098-0121 |doi=10.1103/PhysRevB.74.012403 |bibcode = 2006PhRvB..74a2403R }}</ref> magnetic shape memory materials,<ref name="LakhaniBanerjee2012">{{cite journal |last1=Lakhani |first1=Archana |last2=Banerjee |first2=A. |last3=Chaddah |first3=P. |last4=Chen |first4=X. |last5=Ramanujan |first5=R. V. |title=Magnetic glass in shape memory alloy: Ni<sub>45</sub>Co<sub>5</sub>Mn<sub>38</sub>Sn<sub>12</sub> |journal=Journal of Physics: Condensed Matter |volume=24 |issue=38 |year=2012 |page=386004 |issn=0953-8984 |doi=10.1088/0953-8984/24/38/386004 |pmid=22927562 |arxiv = 1206.2024 |bibcode = 2012JPCM...24L6004L |s2cid=206037831 }}</ref> and other materials.<ref name="KushwahaLakhani2009">{{cite journal |last1=Kushwaha |first1=Pallavi |last2=Lakhani |first2=Archana |last3=Rawat |first3=R. |last4=Chaddah |first4=P. |title=Low-temperature study of field-induced antiferromagnetic-ferromagnetic transition in Pd-doped Fe-Rh |journal=Physical Review B |volume=80 |issue=17 |page=174413 |year=2009 |issn=1098-0121 |doi=10.1103/PhysRevB.80.174413 |arxiv=0911.4552 |bibcode=2009PhRvB..80q4413K|s2cid=119165221 }}</ref>
The interesting feature of these observations of ''T''<sub>g</sub> falling within the temperature range over which the transition occurs is that the first-order magnetic transition is influenced by magnetic field, just like the structural transition is influenced by pressure. The relative ease with which magnetic fields can be controlled, in contrast to pressure, raises the possibility that one can study the interplay between ''T''<sub>g</sub> and ''T''<sub>c</sub> in an exhaustive way. Phase coexistence across first-order magnetic transitions will then enable the resolution of outstanding issues in understanding glasses.

===Critical points===
In any system containing liquid and gaseous phases, there exists a special combination of pressure and temperature, known as the [[Critical point (thermodynamics)|critical point]], at which the transition between liquid and gas becomes a second-order transition. Near the critical point, the fluid is sufficiently hot and compressed that the distinction between the liquid and gaseous phases is almost non-existent. This is associated with the phenomenon of [[critical opalescence]], a milky appearance of the liquid due to density fluctuations at all possible wavelengths (including those of visible light).

===Symmetry===
Phase transitions often involve a [[symmetry breaking]] process. For instance, the cooling of a fluid into a [[crystalline solid]] breaks continuous [[translation symmetry]]: each point in the fluid has the same properties, but each point in a crystal does not have the same properties (unless the points are chosen from the lattice points of the crystal lattice). Typically, the high-temperature phase contains more symmetries than the low-temperature phase due to [[spontaneous symmetry breaking]], with the exception of certain [[accidental symmetry|accidental symmetries]] (e.g. the formation of heavy [[virtual particles]], which only occurs at low temperatures).<ref>{{cite book|last1=Ivancevic|first1=Vladimir G.|last2=Ivancevic|first2=Tijiana, T.|title=Complex Nonlinearity|date=2008|publisher=Springer|location=Berlin|isbn=978-3-540-79357-1|pages=176–177|url=https://books.google.com/books?id=wpsPgHgtxEYC&pg=PA177 |access-date=12 October 2014}}</ref>

===Order parameters<!--'Order parameter' and 'Order parameters' redirect here-->===
An '''order parameter'''<!--boldface per WP:R#PLA--> is a measure of the degree of order across the boundaries in a phase transition system; it normally ranges between zero in one phase (usually above the critical point) and nonzero in the other.<ref>{{cite journal |last1=Clark |first1=J.B. |last2=Hastie |first2=J.W. |last3=Kihlborg |first3=L.H.E. |last4=Metselaar |first4=R. |last5=Thackeray |first5=M.M. |title=Definitions of terms relating to phase transitions of the solid state |journal=Pure and Applied Chemistry |date=1994 |volume=66 |issue=3 |pages=577–594 |doi=10.1351/pac199466030577 |s2cid=95616565 |doi-access=free }}</ref> At the critical point, the order parameter [[susceptibility (disambiguation)|susceptibility]] will usually diverge.

An example of an order parameter is the net [[magnetization]] in a [[ferromagnetic]] system undergoing a phase transition. For liquid/gas transitions, the order parameter is the difference of the densities.

From a theoretical perspective, order parameters arise from symmetry breaking. When this happens, one needs to introduce one or more extra variables to describe the state of the system. For example, in the [[ferromagnetic]] phase, one must provide the net [[magnetization]], whose direction was spontaneously chosen when the system cooled below the [[Curie point]]. However, note that order parameters can also be defined for non-symmetry-breaking transitions.{{cn|date=December 2023}}

Some phase transitions, such as [[superconductivity|superconducting]] and ferromagnetic, can have order parameters for more than one degree of freedom. In such phases, the order parameter may take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition.{{citation needed|date=August 2022}}

There also exist dual descriptions of phase transitions in terms of disorder parameters. These indicate the presence of line-like excitations such as [[Quantum vortex|vortex]]- or [[Topological defect|defect]] lines.

===Relevance in cosmology===
Symmetry-breaking phase transitions play an important role in [[physical cosmology|cosmology]]. As the universe expanded and cooled, the vacuum underwent a series of symmetry-breaking phase transitions. For example, the electroweak transition broke the SU(2)×U(1) symmetry of the [[electroweak force|electroweak field]] into the U(1) symmetry of the present-day [[electromagnetic field]]. This transition is important to explain the asymmetry between the amount of matter and antimatter in the present-day universe, according to  [[electroweak baryogenesis]] theory.

Progressive phase transitions in an expanding universe are implicated in the development of order in the universe, as is illustrated by the work of [[Eric Chaisson]]<ref>{{cite book|last = Chaisson|first = Eric J.|title =Cosmic Evolution|url = https://archive.org/details/cosmicevolutionr00chai|url-access = registration|publisher= Harvard University Press|date = 2001|isbn = 978-0-674-00342-2}}</ref> and [[David Layzer]].<ref>David Layzer, ''Cosmogenesis, The Development of Order in the Universe'', Oxford Univ. Press, 1991</ref>

See also [[relational order theories]] and [[order and disorder]].

===Critical exponents and universality classes===
{{main|critical exponent}}
Continuous phase transitions are easier to study than first-order transitions due to the absence of [[latent heat]], and they have been discovered to have many interesting properties. The phenomena associated with continuous phase transitions are called critical phenomena, due to their association with critical points.

Continuous phase transitions can be characterized by parameters known as [[critical exponent]]s. The most important one is perhaps the exponent describing the divergence of the thermal [[correlation length]] by approaching the transition. For instance, let us examine the behavior of the [[heat capacity]] near such a transition. We vary the temperature ''T'' of the system while keeping all the other thermodynamic variables fixed and find that the transition occurs at some critical temperature ''T''<sub>c</sub>. When ''T'' is near ''T''<sub>c</sub>, the heat capacity ''C'' typically has a [[power law]] behavior:
: <math>C \propto |T_\text{c} - T|^{-\alpha}.</math>

The heat capacity of amorphous materials has such a behaviour near the glass transition temperature where the universal critical exponent ''α'' = 0.59<ref>{{cite journal |url= https://eprints.whiterose.ac.uk/1958/1/ojovanmi1_Topologically2.pdf |doi=10.1088/0953-8984/18/50/007 |bibcode = 2006JPCM...1811507O |title=Topologically disordered systems at the glass transition |journal=Journal of Physics: Condensed Matter |volume=18 |issue=50 |pages=11507–11520 |year=2006 |last1=Ojovan |first1=Michael I. |last2=Lee |first2=William E.|s2cid=96326822 }}</ref> A similar behavior, but with the exponent ''ν'' instead of ''α'', applies for the correlation length.

The exponent ''ν'' is positive. This is different with ''α''. Its actual value depends on the type of phase transition we are considering.

The critical exponents are not necessarily the same above and below the critical temperature. When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then some exponents (such as <math>\gamma</math>, the exponent of the susceptibility) are not identical.<ref>{{cite journal |last1=Leonard |first1=F. |last2=Delamotte |first2=B. |year = 2015 |title=Critical exponents can be different on the two sides of a transition | journal = Phys. Rev. Lett. | volume = 115 | issue = 20| page = 200601 | arxiv = 1508.07852 |bibcode = 2015PhRvL.115t0601L | doi = 10.1103/PhysRevLett.115.200601 |pmid=26613426|s2cid=22181730 }}</ref>

For −1 &lt; ''α'' &lt; 0, the heat capacity has a "kink" at the transition temperature. This is the behavior of liquid helium at the [[lambda transition]] from a normal state to the [[superfluid]] state, for which experiments have found ''α'' = −0.013 ± 0.003.
At least one experiment was performed in the zero-gravity conditions of an orbiting satellite to minimize pressure differences in the sample.<ref>{{cite journal | doi=10.1103/PhysRevB.68.174518 | title=Specific heat of liquid helium in zero gravity very near the lambda point | year=2003 | last1=Lipa | first1=J. | last2=Nissen | first2=J. | last3=Stricker | first3=D. | last4=Swanson | first4=D. | last5=Chui | first5=T. | journal=Physical Review B | volume=68 | issue=17| page=174518 |arxiv = cond-mat/0310163 |bibcode = 2003PhRvB..68q4518L | s2cid=55646571 }}</ref> This experimental value of α agrees with theoretical predictions based on [[variational perturbation theory]].<ref>{{cite journal | doi=10.1103/PhysRevD.60.085001 | title=Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions | year=1999 | last1=Kleinert | first1=Hagen | journal=Physical Review D | volume=60 | issue=8| page=085001 |arxiv = hep-th/9812197 |bibcode = 1999PhRvD..60h5001K | s2cid=117436273 }}</ref>

For 0 &lt; ''α'' &lt; 1, the heat capacity diverges at the transition temperature (though, since ''α'' &lt; 1, the enthalpy stays finite). An example of such behavior is the 3D ferromagnetic phase transition. In the three-dimensional [[Ising model]] for uniaxial magnets, detailed theoretical studies have yielded the exponent ''α'' ≈ +0.110.

Some model systems do not obey a power-law behavior. For example, mean field theory predicts a finite discontinuity of the heat capacity at the transition temperature, and the two-dimensional Ising model has a [[logarithm]]ic divergence. However, these systems are limiting cases and an exception to the rule. Real phase transitions exhibit power-law behavior.

Several other critical exponents, ''β'', ''γ'', ''δ'', ''ν'', and ''η'', are defined, examining the power law behavior of a measurable physical quantity near the phase transition. Exponents are related by scaling relations, such as
: <math>\beta = \gamma / (\delta - 1),\quad \nu = \gamma / (2 - \eta).</math>
It can be shown that there are only two independent exponents, e.g. ''ν'' and ''η''.

It is a remarkable fact that phase transitions arising in different systems often possess the same set of critical exponents. This phenomenon is known as ''universality''. For example, the critical exponents at the liquid–gas critical point have been found to be independent of the chemical composition of the fluid.

More impressively, but understandably from above, they are an exact match for the critical exponents of the ferromagnetic phase transition in uniaxial magnets. Such systems are said to be in the same universality class. Universality is a prediction of the [[renormalization group]] theory of phase transitions, which states that the thermodynamic properties of a system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and are insensitive to the underlying microscopic properties of the system. Again, the divergence of the correlation length is the essential point.

===Critical phenomena===
There are also other critical phenomena; e.g., besides ''static functions'' there is also ''critical dynamics''. As a consequence, at a phase transition one may observe ''critical slowing down'' or ''speeding up''. Connected to the previous phenomenon is also the phenomenon of ''enhanced fluctuations'' before the phase transition, as a consequence of lower degree of stability of the initial phase of the system. The large ''static universality classes'' of a continuous phase transition split into smaller ''dynamic universality'' classes. In addition to the critical exponents, there are also universal relations for certain static or dynamic functions of the magnetic fields and temperature differences from the critical value.{{Citation needed|date=November 2023}}