Editing Phase transition (section)

===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.