Statistical mechanics of a one-dimensional lattice gas

Abstract. We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result of vA ~ Hov] ~ we show tha~, for a large class of interactions, uch a system has no phase transition. The equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be a K-system. 1. Introduct ion and Statement of Results Let 72 be the set of all integers <> 0. We th ink of the elements of 7Z as the sites of a one-dimensional lattice, each site may be occupied by 0 or 1 particle. I f n particles are present on the lattice, at posit ions il < " • • < in, we associate to them a "potent ia l energy" v({il..... i,}) = X X ~(A....,i~). (1.1) ~i {i...... i~} c{i...... i~} The "k-body potent ia l " ~bk is a real function of its arguments]1< " " " < Jk and is assumed to be trans]at ional ly invar iant i.e., if 1 E Z, ~5~(] 1 + 1,...,] ~ + l) = Ck(] 1.....]k). (1.2) Let S (Z and K s be the product of one copy of the set K = {0, 1} for each point of S; K s is the space of all configurations of occupied and empty sites in S; K s is compact for the product of the discrete topologies of the sets {0, 1}. Let (~(K s) be the Banach space of real continuous flmct, ions on K s with the uniform norm and ~(K z) i ts dual, i.e. the space of real measures on K s. I f S C T ( Z we may write K T = K S x K T\s (1.3) and there is a canonical mapping ~TZ: ~(Ks) ~ (d~(KT) such that o:Ts qJ (xs, xT\s) = ~v (xs). (1.4) We denote by a*T the adjoint of aTz: z¢ ~ r/x (q0) = ~t (aTS?) • (1.5