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

:$f\; =\; -kT\; lim\_\{N\; ightarrow\; infty\}\; frac\{1\}\{N\}log\; Z\_N$

The magnetization "M" per site in the thermodynamic limit, depending on the external magnetic field "H" and temperature "T" is given by

:$M(T,H)\; stackrel\{mathrm\{def\{=\}\; lim\_\{N\; ightarrow\; infty\}\; frac\{1\}\{N\}\; left(\; sum\_i\; sigma\_i\; ight)\; =\; -\; left(\; frac\{partial\; f\}\{partial\; H\}\; ight)\_T$

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

:$chi\_T(T,H)\; =\; left(\; frac\{partial\; M\}\{partial\; H\}\; ight)\_T$

and

:$c\_H\; =\; -T\; left(\; frac\{partial^2\; f\}\{partial\; T^2\}\; ight)\_H.$

Definitions

The critical exponents and $delta$ are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

:

:$M(0,H)\; simeq\; |H|^\{1/\; delta\}\; operatorname\{sign\}(H)mbox\{\; for\; \}H\; ightarrow\; 0$

:

:

where

:$t\; stackrel\{mathrm\{def\{=\}\; frac\{T-T\_c\}\{T\_c\}$

measures the temperature relative to the critical point.

Derivation

For the magnetic analogue of the Maxwell relations for the response functions, the relation

:$chi\_T\; (c\_H\; -c\_M)\; =\; T\; left(\; frac\{partial\; M\}\{partial\; T\}\; ight)\_H^2$

follows, and with thermodynamic stability requiring that $c\_h,\; c\_Mmbox\{\; and\; \}chi\_T\; geq\; 0$, one has

:$c\_H\; geq\; frac\{T\}\{chi\_T\}\; left(\; frac\{partial\; M\}\{partial\; T\}\; ight)\_H^2$

which, under the conditions 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.