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


 * $$ M(T,H) \ \stackrel{\mathrm{def}}{=}\  \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) $$

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 $$ \alpha, \alpha', \beta, \gamma, \gamma' $$ 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
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 $$ c_H, c_M\mbox{ and }\chi_T \geq 0 $$, 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.