ABSTRACT
This chapter explores the basic notions of phase transitions, and the manner in which non-analytic thermodynamic behaviour may occur in principle. It presents some concrete calculations of phase transition behaviour in magnetic systems. The chapter is concerned with the transfer matrix method, first introduced by Hendrik Kramers, which reduces the problem of calculating the partition function to the problem of finding the eigenvalues of a certain matrix. It describes the low temperature expansion and Weiss’ mean field theory for magnets. Mean field theory is the simplest treatment of an interacting statistical mechanical system, apart from the approximation of ignoring the interactions all together. The chapter provides a simple, non-systematic derivation of mean field theory for the Ising model. This mean field theory can be easily improved by applying it to clusters of spins rather than individual spins.
