Critical exponents and universality classes Phase transition

continuous phase transitions easier study first-order transitions due absence of latent heat, , have been discovered have many interesting properties. phenomena associated continuous phase transitions called critical phenomena, due association critical points.

it turns out continuous phase transitions can characterized parameters known critical exponents. important 1 perhaps exponent describing divergence of thermal correlation length approaching transition. instance, let examine behavior of heat capacity near such transition. vary temperature t of system while keeping other thermodynamic variables fixed, , find transition occurs @ critical temperature tc . when t near tc , heat capacity c typically has power law behavior,

c
∝

|


t

c


−
t


|


−
α


.


{\displaystyle c\propto |t_{c}-t|^{-\alpha }.}

the heat capacity of amorphous materials has such behaviour near glass transition temperature universal critical exponent α = 0.59 similar behavior, exponent ν instead of α, applies correlation length.

the exponent ν positive. different α. actual value depends on type of phase transition considering.

it believed critical exponents same above , below critical temperature. has been shown not true: when continuous symmetry explicitly broken down discrete symmetry irrelevant (in renormalization group sense) anisotropies, exponents (such



γ


{\displaystyle \gamma }

, exponent of susceptibility) not identical.

for −1 < α < 0, heat capacity has kink @ transition temperature. behavior of liquid helium @ lambda transition normal state superfluid state, experiments have found α = -0.013±0.003. @ least 1 experiment performed in zero-gravity conditions of orbiting satellite minimize pressure differences in sample. experimental value of α agrees theoretical predictions based on variational perturbation theory.

for 0 < α < 1, heat capacity diverges @ transition temperature (though, since α < 1, enthalpy stays finite). example of such behavior 3d ferromagnetic phase transition. in three-dimensional ising model uniaxial magnets, detailed theoretical studies have yielded exponent α ∼ +0.110.

some model systems not obey power-law behavior. example, mean field theory predicts finite discontinuity of heat capacity @ transition temperature, , two-dimensional ising model has logarithmic divergence. however, these systems limiting cases , exception rule. real phase transitions exhibit power-law behavior.

several other critical exponents, β, γ, δ, ν, , η, defined, examining power law behavior of measurable physical quantity near phase transition. exponents related scaling relations, such as

β
=
γ

/

(
δ
−
1
)
,

ν
=
γ

/

(
2
−
η
)


{\displaystyle \beta =\gamma /(\delta -1),\qquad \nu =\gamma /(2-\eta )}

.

it can shown there 2 independent exponents, e.g. ν , η.

it remarkable fact phase transitions arising in different systems possess same set of critical exponents. phenomenon known universality. example, critical exponents @ liquid–gas critical point have been found independent of chemical composition of fluid.

more impressively, understandably above, exact match critical exponents of ferromagnetic phase transition in uniaxial magnets. such systems said in same universality class. universality prediction of renormalization group theory of phase transitions, states thermodynamic properties of system near phase transition depend on small number of features, such dimensionality , symmetry, , insensitive underlying microscopic properties of system. again, divergence of correlation length essential point.