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 = − k T lim N → ∞ 1 N log Z N {\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
M ( T , H ) = d e f lim N → ∞ 1 N ( ∑ i σ i ) {\displaystyle M(T,H)\ {\stackrel {\mathrm {def} }{=}}\ \lim _{N\rightarrow \infty }{\frac {1}{N}}\left(\sum _{i}\sigma _{i}\right)}
where σ i {\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
χ T ( T , H ) = ( ∂ M ∂ H ) T {\displaystyle \chi _{T}(T,H)=\left({\frac {\partial M}{\partial H}}\right)_{T}}
and
c H = T ( ∂ S ∂ T ) H . {\displaystyle c_{H}=T\left({\frac {\partial S}{\partial T}}\right)_{H}.}
Additionally,
c M = + T ( ∂ S ∂ T ) M . {\displaystyle c_{M}=+T\left({\frac {\partial S}{\partial T}}\right)_{M}.}
Definitions The critical exponents α , α ′ , β , γ , γ ′ {\displaystyle \alpha ,\alpha ',\beta ,\gamma ,\gamma '} and δ {\displaystyle \delta } are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows
M ( t , 0 ) ≃ ( − t ) β for t ↑ 0 {\displaystyle M(t,0)\simeq (-t)^{\beta }{\mbox{ for }}t\uparrow 0}
M ( 0 , H ) ≃ | H | 1 / δ sign ( H ) for H → 0 {\displaystyle M(0,H)\simeq |H|^{1/\delta }\operatorname {sign} (H){\mbox{ for }}H\rightarrow 0}
χ T ( t , 0 ) ≃ { ( t ) − γ , for t ↓ 0 ( − t ) − γ ′ , for t ↑ 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}}}
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