1549380323-Statistical Mechanics Theory and Molecular Simulation

One-dimensional Ising model 623

We are interested in the behavior of eqn. (16.5.15) along the criticalisotherm, where
m 0 →0 near the inflection point. Using the expansion tanh−^1 (x)≈x+x^3 /3 +···
gives

h≈kT

[

m+

m^3

3

]

−mJ

=mk

(

T−

J

k

)

+

kT

3

m^3

=mk(T−Tc) +

kT

3

m^3. (16.5.16)

Thus, along the critical isotherm,T=Tc, we find thath∼m^3 , which implies that
δ= 3.
In order to calculateγ, we examine the susceptibilityχasT→Tcfrom above. By
definition,

χ=

∂m

∂h

=

1

∂h/∂m

. (16.5.17)

From eqn. (16.5.16),
∂h
∂m

=k(T−Tc) +kTm^2. (16.5.18)

ForT > Tc,m= 0, hence∂h/∂m∼(T−Tc). Therefore,χ∼(T−Tc)−^1 , from which
it is clear thatγ= 1.
Finally, the exponentαis determined by the behavior of the heat capacityChas
T→Tcfrom above. SinceChis derived from the Gibbs free energy, consider the limit
of eqn. (16.5.8) forT > Tc, asm→0 ath= 0:

G(N,h,T) =−NkTln 2. (16.5.19)

From this expression, it follows thatCh= 0; since, there is no divergence inCh, we
conclude thatα= 0.
In summary, we find that the mean-field exponents for the magnetic model are
α= 0,β= 1/2,γ= 1, andδ= 3, which are exactly the exponents we obtained for
the liquid–gas critical point using the van der Waals equation. Thus, within the mean-
field theory approximation, two very different physical models yield the same critical
exponents, thus providing a concrete illustration of the universality concept. As noted
in Section 4.7, the experimental values of these exponents areα= 0.1,β= 0.34,
γ = 1.35, andδ = 4.2, which shows that mean-field theory is not quantitatively
accurate. Qualitatively, however, mean-field theory reveals manyimportant features
of critical-point behavior (even if it misses the divergence in the heatcapacity) and is,
therefore, a useful first approach.
In order to move beyond mean-field theory, we require an approach capable of ac-
counting for the neglected spatial correlations. We will first examine the Ising model in
one and two dimensions, where the model can be solved exactly. Following this, we will
present an introduction to scaling theory and the renormalization group methodology.