Introduction

The task of statistical mechanics is to derive macroscopic properties of matter from the laws governing the microscopic actions and interactions of individual particles. The systems that are considered in statistical mechanics are those that consist of a large number (on the order of 1027 particles, say, for the molecules in one liter of air) of subsystems (the molecules). To specify such a system on a microscopic level would require the coordinates of a point in 6N-dimensional space, where N is the number of subsystems (or particles) of the system. Recall that we considered these systems in another context in Section 2.8 and the introductions to Chapters 4 and 5.

A macroscopic description of such a system can be given in terms of relatively few quantities such as energy, volume, specific heat, etc., which are called thermodynamic variables, or functions. The entropy of a system is one such thermodynamic variable. Thermodynamics is a study of the relationships that exist between the various thermodynamic variables, and this subject, from a mathematical perspective, can be completely axiomatized [28]. In particular, the equilibrium states of a system can be described in terms of relatively few thermodynamic variables.

We shall not discuss the known relationships between the entropy of a system in an equilibrium state and the other thermodynamic variables of the state. The reader interested in this topic may consult [28] or [154].