Rushbrooke inequality

In statistical mechanics, the Rushbrooke inequality relates the critical exponents of a magnetic system which exhibits a first-order phase transition in the thermodynamic limit for non-zero temperature T. Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as

$f=-kT\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\log Z_N$

The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by

$$\begin{gathered} M(T,H) \\ \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=} \\ \underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\left(\sum _i{\sigma }_i\right) \end{gathered}$$

where ${\sigma }_i$ is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively

${\chi }_T(T,H)={\left(\frac{\partial M}{\partial H}\right)}_T$

and

$c_H=T{\left(\frac{\partial S}{\partial T}\right)}_H.$

Additionally,

$c_M=+T{\left(\frac{\partial S}{\partial T}\right)}_M.$

Definitions

The critical exponents $\alpha ,{\alpha }^{\prime },\beta ,\gamma ,{\gamma }^{\prime }$ and $\delta$ are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

$M(t,0)\simeq (-t{)}^{\beta }\text{ for }t\uparrow 0$

$M(0,H)\simeq |H{|}^{1/\delta }\mathrm{sign}(H)\text{ for }H\to 0$

$$\begin{gathered} {\chi }_T(t,0)\simeq \{\begin{array}{ll}(t{)}^{-\gamma },& \text{<mrow data-mjx-texclass="ORD">for</mrow>} \\ t\downarrow 0\\ (-t{)}^{-{\gamma }^{\prime }},& \text{<mrow data-mjx-texclass="ORD">for</mrow>} \\ t\uparrow 0\end{array} \end{gathered}$$

$$\begin{gathered} c_H(t,0)\simeq \{\begin{array}{ll}(t{)}^{-\alpha }& \text{<mrow data-mjx-texclass="ORD">for</mrow>} \\ t\downarrow 0\\ (-t{)}^{-{\alpha }^{\prime }}& \text{<mrow data-mjx-texclass="ORD">for</mrow>} \\ t\uparrow 0\end{array} \end{gathered}$$

where

$$\begin{gathered} t \\ \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=} \\ \frac{T-T_c}{T_c} \end{gathered}$$

measures the temperature relative to the critical point.

Derivation

Using the magnetic analogue of the Maxwell relations for the response functions, the relation

${\chi }_T(c_H-c_M)=T{\left(\frac{\partial M}{\partial T}\right)}_H^2$

follows, and with thermodynamic stability requiring that $c_H,c_M\text{ and }{\chi }_T\ge 0$, one has

$c_H\ge \frac{T}{{\chi }_T}{\left(\frac{\partial M}{\partial T}\right)}_H^2$

which, under the conditions $H=0,t>0$ and the definition of the critical exponents gives

$(-t{)}^{-{\alpha }^{\prime }}\ge \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\cdot (-t{)}^{{\gamma }^{\prime }}(-t{)}^{2(\beta -1)}$

which gives the Rushbrooke inequality

${\alpha }^{\prime }+2\beta +{\gamma }^{\prime }\ge 2.$

Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.