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: \(\gamma\)

Susceptibility

\[\left. \chi \right. = \chi_0 \epsilon^{-\gamma}\]

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.