Critical exponents

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \alpha =0.1105_{-0.027}^{+0.025}} .

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma =1.3177(5)} for the three-dimensional XY universality class.

Correlation length

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

Theoretically one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu = 0.63012(16)} for the three dimensional Ising model, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu = 0.67155(27)} for the three-dimensional XY universality class.

Inequalities

Fisher inequality

The Fisher inequality (Eq. 5 )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma \le (2-\eta) \nu}

Griffiths inequality

The Griffiths inequality (Eq. 3 ):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1+\delta)\beta \ge 2-\alpha'}

Josephson inequality

The Josephson inequality

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha' + 2\beta + \gamma' \ge 2} .

Using the above-mentioned values one has:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.1096 + (2\times0.32653) + 1.2373 = 1.99996}

Widom inequality

The Widom inequality

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma' \ge \beta(\delta -1)}

Gamma divergence

When approaching the critical point along the critical isochore (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T > T_c} ) the divergence is of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \right. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \right. \kappa_T \sim (p-p_c)^{-\epsilon}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} is 2/3 for the Van der Waals equation of state, and is usually 0.75 to 0.8.

See also

• Universality classes