Hosta

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)

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 S}{\partial T}}\right)_{H}.

Additionally,

c_{M}=+T\left({\frac {\partial S}{\partial T}}\right)_{M}.

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