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Abstract. We say that a representation of an algebra, of local observables has short-range correlations if any observable which can be measured outside all bounded sets is a multiple of the identity, and that a state has finite range correlations if the corresponding cyclic representation does. We characterize states with short-range correlations by a cluster property. For classical lattice systems and continuous systems with hard cores, we give a definition of equilibrium state for a specific interaction, based on a local version of the grand canonical prescription; an equilibrium state need not be translation invariant. We show that every equi. librium state has a unique decomposition into equilibrium states with short-range correlations. We use the properties of equilibrium states to prove some negative results about the existence of metastable states. We show that the correlation functions for an equilibrium state satisfy the Kirkwood-Salsburg equations; thus, at low activity, there is only one equilibrium state for a given interaction, temperature, and chemical potential. Finally, we argue heuristically that equilibrium states are invariant under time-evolution.

A. THERMODYNAMIC LIMITS AI. INTRODUCTION AND PRELIMINARIESThe objective of statistical mechanics is to explain the macroscopic properties of matter on the basis of the behavior of the atom and molecules of which it is composed.One of the most striking facts about macroscopic matter is that in spite of being fantastically complicated on the atomic level--to specify the positions and velocities of all molecules in a glass of water would mean specifying something of the order of 1025 parameters--its macroscopic behavior is describable in terms of a very small number of parameters, e.g., the temperature and density for a system containing only one kind of molecule. We will begin these lectures by describing an explanation-scheme to account for this fact. The term explanation-scheme is intended to convey the fact that what is outlined here is not a finished piece of work--there are many difficult theorems to be proved before we can be sure that the proposed explanation is correct, and it will surely be necessary to modify the technical details of the formulation--but rather a set of ideas which seem to me to offer the most promising avenue for understanding the effectiveness of statistical mechanics for computing the properties of matter. My discussion will draw heavily on ideas developed by Ruelle in an important but little known paper [i0].The explanation I want to consider is based on two main ideas:(a) We look only at observables of a special kind; roughly speaking, those which test correlations between particles which are not too far apart. This description is a little obscure; we will give a more precise one in a moment.(b) We consider the behavior of the system as the number of particles becomes very large.The spirit of our approach is to try to argue that, for a very large system, all observables are "essentially" determined by the energy and the density. By this we mean that, if we consider systems of fixed density but of increasing size, and hence of increasing number of particles, the probability distribution of each observable (approximately normalized) with respect to Lebesgue measure on each energy surface,

We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).