Statistical Mechanics - Physics at Oregon State University

Chapter 9
General methods: critical
exponents.
9.1 Introduction.
In this chapter we will consider a variety of methods that can give us approximations
in statistical mechanics. In the previous chapter we looked at the mean
field approximation and its cousins. We found that mean field gives a reasonable
description if there is a phase transition. It also leads to a decent estimate of
the transition temperature, except for the one dimensional Ising chain. Cluster
approximations improve the estimates of the transition temperature. All
in all these methods are very useful to describe phase transitions and help us
understand different models. Where they fail, however, is in providing values
for critical exponents. They always give the same values, no matter what the
model is, due to the mean field nature. This is easy to understand. In a mean
field model we approximate effects further away by an average. When correlation
lengths become large, a cluster will always sample that average value
and essentially see a mean field. So on this chapter we focus in more detail on
correlation functions and critical exponents.
There are two possible goals of calculations in statistical mechanics. One
useful result is to be able to find certain thermal averages of quantities. But
the real goal is to find the appropriate free energy, since everything follows
from there. That second goal is more difficult. For example, in the Bethe
approximation we needed to include a difficult term f(h ′ ) in the energy. For
the calculation of averages like 〈σ0σj〉 that term dropped out. But without that
term we cannot evaluate free energies.
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