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

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

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

${\displaystyle c_{H}=T\left({\frac {\partial S}{\partial T))\right)_{H}.}$

Additionally,

${\displaystyle c_{M}=+T\left({\frac {\partial S}{\partial T))\right)_{M}.}$

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

Using 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.