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

$f=-kT\lim _{N\rightarrow \infty }{\frac {1}{N}}\log Z_{N}$

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

$M(T,H)\ {\stackrel {\mathrm {def} }{=}}\ \lim _{N\rightarrow \infty }{\frac {1}{N}}\left(\sum _{i}\sigma _{i}\right)=-\left({\frac {\partial f}{\partial H}}\right)_{T}$

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

$c_{H}=-T\left({\frac {\partial ^{2}f}{\partial T^{2}}}\right)_{H}.$

Definitions

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

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

$M(0,H)\simeq |H|^{1/\delta }\operatorname {sign} (H){\mbox{ for }}H\rightarrow 0$

$\chi _{T}(t,0)\simeq {\begin{cases}(t)^{-\gamma },&{\textrm {for}}\ t\downarrow 0\\(-t)^{-\gamma '},&{\textrm {for}}\ t\uparrow 0\end{cases}}$

$c_{H}(t,0)\simeq {\begin{cases}(t)^{-\alpha }&{\textrm {for}}\ t\downarrow 0\\(-t)^{-\alpha '}&{\textrm {for}}\ t\uparrow 0\end{cases}}$

where

$t\ {\stackrel {\mathrm {def} }{=}}\ {\frac {T-T_{c}}{T_{c}}}$

measures the temperature relative to the critical point.

Derivation

For 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 $c_{h},c_{M}{\mbox{ and }}\chi _{T}\geq 0$, one has

$c_{H}\geq {\frac {T}{\chi _{T}}}\left({\frac {\partial M}{\partial T}}\right)_{H}^{2}$

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

$(-t)^{-\alpha '}\geq \mathrm {constant} \cdot (-t)^{\gamma '}(-t)^{2(\beta -1)}$

which gives the Rushbrooke inequality

$\alpha '+2\beta +\gamma '\geq 2.$

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