Physics: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

[math]\displaystyle{ f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N }[/math]

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

[math]\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 }[/math]

where [math]\displaystyle{ \sigma_i }[/math] is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively

[math]\displaystyle{ \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T }[/math]

and

[math]\displaystyle{ c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H. }[/math]

Definitions

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

[math]\displaystyle{ M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 }[/math]

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

[math]\displaystyle{ \chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases} }[/math]

[math]\displaystyle{ c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases} }[/math]

where

[math]\displaystyle{ t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c} }[/math]

measures the temperature relative to the critical point.

Derivation

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

[math]\displaystyle{ \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 }[/math]

follows, and with thermodynamic stability requiring that [math]\displaystyle{ c_H, c_M\mbox{ and }\chi_T \geq 0 }[/math], one has

[math]\displaystyle{ c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 }[/math]

which, under the conditions [math]\displaystyle{ H=0, t\gt 0 }[/math] and the definition of the critical exponents gives

[math]\displaystyle{ (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} }[/math]

which gives the Rushbrooke inequality

[math]\displaystyle{ \alpha' + 2\beta + \gamma' \geq 2. }[/math]

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