Statistical Mechanics - Physics at Oregon State University

Chapter 8
Mean Field Theory: critical
temperature.
8.1 Introduction.
When are systems interesting?
In the previous chapters we have developed the principles of statistical mechanics.
Most of the applications were to simple models. The only important
system we discussed in detail was a gas of independent particles. The model we
discussed for that system is very important, since a gas is almost always close
to ideal. Quantum effects can be explained in this model in a qualitative way.
The theoretical model of an ideal gas is also a very good starting point for the
treatment of a real gas, as we will see in a next chapter.
In this chapter we discuss systems of real interest. By that we mean systems
that show a phase transition from one state to another. Real systems, of course,
are quite complex, so we need to find a simple model that applies to realistic
systems, yet can be approximately solved by analytical means. Therefore, we
focus on a theoretical model which applies to many effects encountered in solids.
The most important characteristic of a solid is the fact that the atomic
positions are fixed, if we ignore the small fluctuations due to phonons. With
each atomic position we can associate other degrees of freedom, like spin or
atom-type. Solid state physics is concerned with the procedures of obtaining
information pertaining to these degrees of freedom, experimentally as well as
theoretically.
Theoretical models of internal degrees of freedom coupled to fixed lattice
sites are called lattice models. These models play an important role in many
areas of physics and can be generalized to cases where particles are moving and
are not restricted to specific lattice sites. Here we will only discuss the most
elementary member of the class of lattice models, the Ising model. We will use
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