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# Critical exponents

Critical exponents. Groups of critical exponents form universality classes.

## Reduced distance: $\epsilon$ $\epsilon$ is the reduced distance from the critical temperature, i.e. $\epsilon = \left| 1 -\frac{T}{T_c}\right|$

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: $\alpha$

The isochoric heat capacity is given by $C_v$ $\left. C_v\right.=C_0 \epsilon^{-\alpha}$

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

## Magnetic order parameter exponent: $\beta$

The magnetic order parameter, $m$ is given by $\left. m\right. = m_0 \epsilon^\beta$

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

## Susceptibility exponent: $\gamma$ $\left. \chi \right. = \chi_0 \epsilon^{-\gamma}$

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

## Correlation length $\left. \xi \right.= \xi_0 \epsilon^{-\nu}$

Theoretically one has $\nu = 0.63012(16)$ for the three dimensional Ising model, and $\nu = 0.67155(27)$ for the three-dimensional XY universality class.

## Inequalities

#### Fisher inequality

The Fisher inequality (Eq. 5 ) $\gamma \le (2-\eta) \nu$

#### Griffiths inequality

The Griffiths inequality (Eq. 3 ): $(1+\delta)\beta \ge 2-\alpha'$

#### Josephson inequality

The Josephson inequality $d\nu \ge 2-\alpha$

#### Rushbrooke inequality

The Rushbrooke inequality (Eq. 2 ), based on the work of Essam and Fisher (Eq. 38 ) is given by $\alpha' + 2\beta + \gamma' \ge 2$.

Using the above-mentioned values one has: $0.1096 + (2\times0.32653) + 1.2373 = 1.99996$

#### Widom inequality

The Widom inequality $\gamma' \ge \beta(\delta -1)$

## Gamma divergence

When approaching the critical point along the critical isochore ( $T > T_c$) the divergence is of the form $\left. \right. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}$

where $\kappa_T$ is the isothermal compressibility. $\gamma$ 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 $\left. \right. \kappa_T \sim (p-p_c)^{-\epsilon}$

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