1549380323-Statistical Mechanics Theory and Molecular Simulation

616 Critical phenomena

Table 16.1Comparison of thermodynamic variables and relations between gas–liquid and
magnetic systems.

Quantity Gas–Liquid Magnetic Quantity

Pressure P h Magnetic field

Volume V −M=−Nm Magnetization

Isothermal Magnetic

compressibility κT=−(1/V)(∂P/∂V) χ=∂m/∂h susceptibility

Helmholtz Helmholtz

free energy A(N,V,T) A(N,M,T) free energy

Gibbs Gibbs

free energy G(N,P,T) G(N,h,T) free energy

Pressure Magnetic field

relation P=−∂A/∂V h=∂A/∂M relation

Volume Magnetization

relation V=∂G/∂P M=−∂G/∂h relation

Const. Const.

volume magnetization

heat capacity CV=−T

(

∂^2 A/∂T^2

)

V CM=−T

(

∂^2 A/∂T^2

)

M heat capacity

Const. pressure Const. field

heat capacity CP=−T

(

∂^2 G/∂T^2

)

P Ch=−T

(

∂^2 G/∂T^2

)

h heat capacity

Along the critical isotherm, the behavior of the equation of state near the inflection
point is
h∼|m|δsign(m). (16.3.8)

Finally, asT →Tcfrom below, the discontinuity in the magnetization depends on
temperature according to the power-law

m∼|Tc−T|β. (16.3.9)

In this way, a perfect analogy is established between the magnetic and gas–liquid
systems. Before proceeding to analyze the magnetic system model, however, we first
clarify the concept of universality and provide a definition of universality classes.

16.4 Universality classes

In a perfectly ordered magnetic state, the magnetization per spinmcan take one of
two values,m= 1 orm=−1, depending on the direction in which the spins point.
In the former case,σ 1 = 1,...,σN= 1, while in the latterσ 1 =− 1 ,...,σN=−1. If we
perform a variable transformation

σi′=−σi, (16.4.1)