The purpose of this introduction is to provide motivation and a historical setting for the problems treated by Israel in the following. The historical approach is not inappropriate because some of the questions discussed were also treated by the old masters of thermodynamics and statistical mechanics over the last hundred years; yet they are perennial, arising anew in the context of the statistical mechanics of each dynamical system. The modern theory acquires additional flavor when it is savored as part of this long development.

Here are some examples of such perennial questions.

a) What is the structure of the family...

The Ising model is a simple example of a statistical-mechanical system. At each site of the lattice Z^νwe assume there is a “spin” which can be either “up” (+1) or “down” (−1). Thus for each subset Λ of Z^νWe have the space Ω_Λ= {−1, +1}^Λof configurations in Λ. With the product topology this forms a compact metric space, even if Λ is infinite. The full configuration space Ω_Z^νwill he denoted simply by Ω, andσ_i(ω) will denote the spin at site i ∊ Z^νin the configurationω. The Hamiltonian for the system in a...

In this chapter we will define invariant equilibrium states of our classical and quantum lattice systems in terms of tangent functionals to the pressure on the space${\cal {B}}$of interactions, and show that this is equivalent to an alternative definition by the Variational Principle: invariant equilibrium states for a given interaction maximize the difference between entropy and energy per site (recall that we are absorbing the temperature into the interaction, so thatβ= 1). The first step is to develop the connection between translation-invariant states and functionals on the space of interactions.

A state on a C*-algebra A with...

In Chapter II we defined invariant equilibrium states in terms of tangent functionals to the pressure, and showed that this was equivalent to the formulation in terms of a Variational Principle. An alternative approach is to define (not necessarily invariant) equilibrium states, using the Dobrushin-Lanford-Ruelle (DLR) equations [18] for classical systems and the Kubo-Martin-Schwinger (KMS) boundary conditions [14] for quantum systems. The DLR equations are motivated by the consideration of relative probabilities of configurations in a finite system with fixed boundary conditions. The KMS conditions are rather more subtle, and require consideration of the time evolution of the quantum system....

The invariant equilibrium states for an interaction$\Phi \epsilon {\cal {B}}$form a convex set Δ_Φ, compact in the weak-* topology. Thus it is natural from the mathematical point of view to consider the extreme points of Δ_Φ. Note that these are also extremal in the set E^Iof all invariant states, since by the Variational Principle the invariant equilibrium states are those invariant states on which the affine functionρ↦ s(ρ) −ρ(Α_Φ) attains its supremum. These extreme points turn out to be important from the physical point of view as well, because they represent “pure thermodynamic phases.” The “pure phases”...

A standard type of problem in statistical mechanics is to describe the invariant equilibrium states for a given interaction. In this chapter we consider the reverse situation: given an invariant stateρ, we look for an interaction having an invariant equilibrium state$\tilde{\rho}$which bears some resemblance toρ. In particular, ifρexhibits some type of long-range order, we would like$\tilde{\rho}$to share this property. This approach leads to some very general existence results for phase transitions, showing that a given type of phase transition occurs for some member of a certain class of interactions. In Section V.2,...

By a theorem of Mazur ([19], Satz 2), a continuous convex function on a real separable Banach space has unique tangent functionals on a dense G_δset of points. This was applied to statistical mechanics by Gallavotti and Miracle-Solé [8] and Ruelle [24], yielding a “weak Gibbs Phase Rule” in any Banach space of interactions dense in${\cal {B}}$and containing${\cal {B}}_{0}$(the finite-range interactions), a dense G_δset of interactions have unique invariant equilibrium states. In this chapter we prove a stronger version of the Gibbs Phase Rule, with, in a sense, much of the content of the original “thermodynamic”...