Rushbrooke inequality

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 alpha, alpha', eta, gamma, gamma' and delta are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

: M(t,0) simeq (-t)^{eta}mbox{ for }t uparrow 0

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

: chi_T(t,0) simeq egin{cases} (t)^{-gamma}, & extrm{for} t downarrow 0 \ (-t)^{-gamma'}, & extrm{for} t uparrow 0 end{cases}

: c_H(t,0) simeq egin{cases} (t)^{-alpha} & extrm{for} t downarrow 0 \ (-t)^{-alpha'} & extrm{for} t uparrow 0 end{cases}

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 H=0, t<0 and the definition of the critical exponents gives

: (-t)^{-alpha'} geq mathrm{constant}cdot(-t)^{gamma'}(-t)^{2(eta-1)}

which gives the Rushbrooke inequality

: alpha' + 2eta + gamma' geq 2.

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


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