# Critical exponents

Critical exponents. Groups of critical exponents form universality classes.

## Reduced distance: ${\displaystyle \epsilon }$

${\displaystyle \epsilon }$ is the reduced distance from the critical temperature, i.e.

${\displaystyle \epsilon =\left|1-{\frac {T}{T_{c}}}\right|}$

Note that this implies a certain symmetry when the critical point is approached from either 'above' or 'below', which is not necessarily the case.

## Heat capacity exponent: ${\displaystyle \alpha }$

The isochoric heat capacity is given by ${\displaystyle C_{v}}$

${\displaystyle \left.C_{v}\right.=C_{0}\epsilon ^{-\alpha }}$

Theoretically one has ${\displaystyle \alpha =0.1096(5)}$[1] for the three dimensional Ising model, and ${\displaystyle \alpha =-0.0146(8)}$[2] for the three-dimensional XY universality class. Experimentally ${\displaystyle \alpha =0.1105_{-0.027}^{+0.025}}$[3].

## Magnetic order parameter exponent: ${\displaystyle \beta }$

The magnetic order parameter, ${\displaystyle m}$ is given by

${\displaystyle \left.m\right.=m_{0}\epsilon ^{\beta }}$

Theoretically one has ${\displaystyle \beta =0.32653(10)}$[1] for the three dimensional Ising model, and ${\displaystyle \beta =0.3485(2)}$[2] for the three-dimensional XY universality class.

## Susceptibility exponent: ${\displaystyle \gamma }$

${\displaystyle \left.\chi \right.=\chi _{0}\epsilon ^{-\gamma }}$

Theoretically one has ${\displaystyle \gamma =1.2373(2)}$[1] for the three dimensional Ising model, and ${\displaystyle \gamma =1.3177(5)}$[2] for the three-dimensional XY universality class.

## Correlation length

${\displaystyle \left.\xi \right.=\xi _{0}\epsilon ^{-\nu }}$

Theoretically one has ${\displaystyle \nu =0.63012(16)}$[1] for the three dimensional Ising model, and ${\displaystyle \nu =0.67155(27)}$[2] for the three-dimensional XY universality class.

## Inequalities

#### Fisher inequality

The Fisher inequality (Eq. 5 [4])

${\displaystyle \gamma \leq (2-\eta )\nu }$

#### Griffiths inequality

The Griffiths inequality (Eq. 3 [5]):

${\displaystyle (1+\delta )\beta \geq 2-\alpha '}$

#### Josephson inequality

The Josephson inequality [6][7][8]

${\displaystyle d\nu \geq 2-\alpha }$

#### Rushbrooke inequality

The Rushbrooke inequality (Eq. 2 [10]), based on the work of Essam and Fisher (Eq. 38 [11]) is given by

${\displaystyle \alpha '+2\beta +\gamma '\geq 2}$.

Using the above-mentioned values[1] one has:

${\displaystyle 0.1096+(2\times 0.32653)+1.2373=1.99996}$

#### Widom inequality

The Widom inequality [12]

${\displaystyle \gamma '\geq \beta (\delta -1)}$

## Gamma divergence

When approaching the critical point along the critical isochore (${\displaystyle T>T_{c}}$) the divergence is of the form

${\displaystyle \left.\right.\kappa _{T}\sim (T-T_{c})^{-\gamma }\sim (p-p_{c})^{-\gamma }}$

where ${\displaystyle \kappa _{T}}$ is the isothermal compressibility. ${\displaystyle \gamma }$ is 1.0 for the Van der Waals equation of state, and is usually 1.2 to 1.3.

## Epsilon divergence

When approaching the critical point along the critical isotherm the divergence is of the form

${\displaystyle \left.\right.\kappa _{T}\sim (p-p_{c})^{-\epsilon }}$

where ${\displaystyle \epsilon }$ is 2/3 for the Van der Waals equation of state, and is usually 0.75 to 0.8.