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Rushbrooke inequality

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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.

Contents

Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as

f = − k T 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 )   = d e f   lim N → ∞ 1 N ( ∑ i σ i ) = − ( ∂ f ∂ H ) T

where σ 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

χ T ( T , H ) = ( ∂ M ∂ H ) T

and

c H = − T ( ∂ 2 f ∂ T 2 ) H .

Definitions

The critical exponents α , α ′ , β , γ , γ ′ and δ are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

M ( t , 0 ) ≃ ( − t ) β  for  t ↑ 0 M ( 0 , H ) ≃ | H | 1 / δ sign ⁡ ( H )  for  H → 0 χ T ( t , 0 ) ≃ { ( t ) − γ , for   t ↓ 0 ( − t ) − γ ′ , for   t ↑ 0 c H ( t , 0 ) ≃ { ( t ) − α for   t ↓ 0 ( − t ) − α ′ for   t ↑ 0

where

t   = d e f   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

χ T ( c H − c M ) = T ( ∂ M ∂ T ) H 2

follows, and with thermodynamic stability requiring that c h , c M  and  χ T ≥ 0 , one has

c H ≥ T χ T ( ∂ M ∂ T ) H 2

which, under the conditions H = 0 , t > 0 and the definition of the critical exponents gives

( − t ) − α ′ ≥ c o n s t a n t ⋅ ( − t ) γ ′ ( − t ) 2 ( β − 1 )

which gives the Rushbrooke inequality

α ′ + 2 β + γ ′ ≥ 2.

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

References

Rushbrooke inequality Wikipedia
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