Rushbrooke inequality

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. ==References==