Equilibrium states, a concept originating from statistical mechanics, are probability measures on topological spaces that are characterized by variational principles. They maximize the sum (or difference) of an entropy and an energy like quantity. Depending on the special choice of the variational problem these measures can have various interesting properties. Some of them are discussed in this introductory chapter on a rather elementary level. Starting from the roots of these ideas in equilibrium statistical mechanics, we outline the connection between equilibrium states and the theory of large deviations and introduce the Ising model of ferromagnetism on a finite lattice as a more concrete example (Sections 1.1 and 1.2). Then we indicate how Markov measures on finite alphabets can be characterized by a variational principle (Section 1.3), and the final section deals with the role played by equilibrium states in the ergodic theory of dynamical systems. That section also furnishes the background for a first discussion of Birkhoff's ergodic theorem.

Equilibrium states in finite systems

A physical system consisting of many particles can be described on two levels: Microscopically it is determined by its configuration, i.e., by the positions and momenta of all particles. Knowing the configuration of a system which obeys the laws of classical mechanics and which is not influenced from outside allows one in principle to determine its exact configuration at any future time. Of course, this fact is of little practical relevance, because the configuration of a realistic large system (e.g., the positions and momenta of all 2.7 · 1022 molecules of an ideal gas in a one litre container) cannot even approximately be known.