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.

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