Condensed matter physics by its very nature deals with systems with a large number of degrees of freedom, i.e. systems for which a statistical description is essential. In this chapter, we will present a rather comprehensive review of the fundamentals of thermodynamics and statistical mechanics. Much of this chapter, especially the parts dealing with homogeneous fluids (ideal and interacting gases and liquids), should be familiar to everyone. They are included here mostly to establish a basis for discussing more complicated ordered systems. As we saw in the preceding chapter, a great deal of useful and experimentally accessible information is contained in correlation functions such as the density-density correlation function Cnn(x,x′). This chapter will define these functions in a statistical mechanical context and develop a method of calculating them using functional differentiation with respect to spatially varying external fields. Functional differentiation will allow us to calculate position-dependent correlation and response functions by a simple generalization of the familiar technique used to calculate the magnetic susceptibility by differentiating the free energy with respect to the magnetic field. This is a very powerful tool that will be used throughout this book. It is presented here first in a familiar context that should make it easy to grasp.

After reviewing the properties of homogeneous fluids, we will introduce order parameters in Sec. 3.5 and show how they modify both thermodynamics and statistical mechanics.