1549380323-Statistical Mechanics Theory and Molecular Simulation

622 Critical phenomena

m m

g(0,T) g(0,T)

m 0 m 0

(a) (b)

Fig. 16.7Free energy of eqn. (16.5.12),g(0,T). (a)T < Tc. (b)T > Tc.

We now turn to the calculation of the critical exponents for the Ising model within
the mean-field theory. From the free energy plot in Fig. 16.7(a), wecan obtain the
exponentβdirectly. Recall thatβdescribes how the discontinuity associated with the
first-order phase transition forT < Tcdepends on temperature asT→Tc, and for
this, we need to know howm 0 depends onTforT < Tc. The dependence ofm 0 onT
is determined by the condition thatg(0,T) be a minimum atm=m 0 :

∂g(0,T)

∂m

∣

∣

∣

∣

m=m 0

= 0, (16.5.13)

or

2 J(1−βJ)m 0 + 4c 2 m^30 = 0

2 J

T

(

T−

J

k

)

+ 4c 2 m^20 = 0

2 J

T

(T−Tc) + 4c 2 m^20 = 0

m 0 ∼(Tc−T)^1 /^2 , (16.5.14)

from which it is clear thatβ= 1/2.
In order to determineδ, the mean-field equation of state is needed, which is pro-
vided by eqn. (16.5.10). Solving eqn. (16.5.10) forhyields

h=kTtanh−^1 (m)−mJ. (16.5.15)