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 = − 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 .