Rushbrooke inequality explained

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} (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{ def }\ \lim_ \frac \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( (\partial M)/(\partial H) \right) _{T}

and

c _{H} = T \left( (\partial S)/(\partial T) \right) _{H.}

Additionally,

c _{M} = + T \left( (\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 followsM(t, 0 ) \simeq ( - t) ^{\beta} for t \uparrow 0 M( 0 ,H) \simeq |H| ^{1/} \operatorname{sign}(H) for H → 0 \chi _{T(t,0)} \simeq \begin{cases} (t) ^{-\gamma} , & rm{for} t \downarrow 0 \\ ( - t) ^{-\gamma'} , & rm{for} t \uparrow 0 \end{cases} c _{H(t,0)} \simeq \begin{cases} (t) ^{-\alpha} & rm{for} t \downarrow 0 \\ ( - t) ^{-\alpha'} & rm{for} t \uparrow 0 \end{cases}

where

t \stackrel{ def }\ \frac

measures the temperature relative to the critical point.

Derivation

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

\chi _{T} (c _{H} - c _{M)} = T \left( (\partial M)/(\partial T) \right) ^{2} _{H}

follows, and with thermodynamic stability requiring that

c _{H,} c _{Mand\chi} _{T} \geq 0 , one hasc _{H} \geq (T)/(\chi _{T}) \left( (\partial M)/(\partial T) \right) ^{2} _{H}

which, under the conditions

H =0 , t> 0 and the definition of the critical exponents gives( - t) ^{-\alpha'} \geq constant ⋅ ( - t) ^{\gamma'} ( - t) ^{2(\beta-1)}

which gives the Rushbrooke inequality

\alpha' + 2 \beta + \gamma' \geq 2 .

Remarkably, in experiment and in exactly solved models, the inequality actually holds as an equality.