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

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

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

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

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

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

and

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

Definitions

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

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

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

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

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

where

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

measures the temperature relative to the critical point.

Derivation

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

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

follows, and with thermodynamic stability requiring that ${\displaystyle c_{H},c_{M}{\mbox{ and }}\chi _{T}\geq 0}$, one has

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

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

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

which gives the Rushbrooke inequality

${\displaystyle \alpha '+2\beta +\gamma '\geq 2.}$

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