1549380323-Statistical Mechanics Theory and Molecular Simulation

618 Critical phenomena

universality class. By contrast, in the Heisenberg model of eqn. (16.4.2), the spins can
point in any spatial direction, and the order parameter is the magnetization vector

M=

〈N

∑

i=1

σi

〉

, (16.4.4)

which has dimensionn= 3, corresponding to the three components ofM. Such a
system could be used to determine the exponents of thed= 3,n= 3 universality
class.
Having established the concept of a universality class, we will now proceed to
analyze the Ising model in order to gain an understanding of systems in thed= 3,
n= 1 universality class near their critical points.

16.5 Mean-field theory

We begin our treatment of the Ising model by invoking an approximation scheme
known as themean-field theory. In this approach, spatial correlations are neglected,
and each particle is assumed to experience an “average” or “mean”field due to the
other particles in the system. Before examining the magnetic system, let us first note
that we previously encountered this approximation in our discussionof the van der
Waals equation in Section 4.7. We derived the van der Waals equation ofstate from
a perturbation expansion of the potential energy of a system, and we obtained the
approximate Helmholtz free energy of eqn. (4.7.24):

A=−

1

β

ln

(

ZN(0)

N!λ^3 N

)

+〈U 1 〉 0 −

β

2

(

〈U 12 〉 0 −〈U 1 〉^20

)

+···,

whereZN^0 is the configurational partition function due to the unperturbed potential
U 0. Note that the second term in the free energy is just the averageof the pertur-
bationU 1 , while the third term is the fluctuation in this potential〈(U 1 −〈U 1 〉)^2 〉. In
the derivation of the van der Waals equation, the fluctuation term was completely
neglected. Furthermore, the unperturbed configurational partition function was taken
to be that of an ideal gas in a reduced volume. Thus, all of the interactions between
particles were assumed to arise fromU 1 , and the approximation of retaining only the
first two terms in the free energy expression amounted to replacingU 1 with〈U 1 〉in
the configurational partition function, i.e.,

ZN=ZN(0)〈e−βU^1 〉≈ZN(0)e−β〈U^1 〉 (16.5.1)

(cf. eqn. (4.7.4)). The mean-field theory approximation can recover the first two terms
in the free energy. Recall that the van der Waals equation, despiteits crudeness,
predicts a gas-to-liquid phase transition as well as a critical point. The four primary
exponents were found in Section 4.7 to beα= 0,β= 1/2,γ= 1, andδ= 3 within the
mean-field approximation. In our discussion of the van der Waals equation, we referred
to the fact that the isotherms are unrealistic forT < Tcowing to regions where bothP
andVincrease simultaneously. In Fig. 4.8, the correction to aT < Tcisotherm, which
appears as the thin solid straight line, is necessary for the calculation of the exponent