1549380323-Statistical Mechanics Theory and Molecular Simulation

Ising model 615

h

T

Tc

M > 0

M < 0

(Spin up)

(Spin down)

M = 0

Fig. 16.4Phase diagram of the Ising model.

m

h

T = T

T > T

T < Tc

c

c

Fig. 16.5Equation of state of the Ising model.

separates these two modes of behavior. Ath= 0, the isotherm has zero curvature,
meaning that∂h/∂m= 0 and∂^2 h/∂m^2 = 0.
Table 16.1 draws an analogy between the gas–liquid and magnetic cases. It lists the
basic thermodynamic variables and thermodynamic relations among these variables,
which will be needed throughout our discussion. Accordingly, the primary critical
exponents are defined as follows. Ath= 0 andm= 0 (the values of the magnetic field
and magnetization at the critical point), asT→Tcfrom above, the heat capacity at
constant magnetizationCMdiverges as

CM∼|T−Tc|−α. (16.3.6)

Similarly, ath= 0 andm= 0, asT→Tcfrom above, the susceptibility diverges as

χ∼|T−Tc|−γ. (16.3.7)