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 The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by where
is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively and Additionally, The critical exponents
are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows
where measures the temperature relative to the critical point.
and the definition of the critical exponents gives which gives the Rushbrooke inequality Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.