Equilibrium states and ergodic properties of infinite systems

We discuss the ergodic theoretic structure of infinite classical systems and present results on the ergodic properties of some simple model systems, e.g., ideal gas, Lorentz gas, Harmonic crystal. (The ergodic properties of the latter system are shown to be related in a simple way to the spectrum of the force matrix; when the spectrum is absolutely continuous, as in the translation-invariant crystal, the flow is Bernoulli.) We argue that ergodic properties, suitably refined by the inclusion of space translations, and other structure, are important for an understanding of nonequilibrium properties of macroscopic systems [1–5]. Possible additional structures include requirements of stability for the stationary state. We shall present results on the classical analog of the work by Haag, Kastler, and Trych-Pohlmeyer [6], Araki [7], and others [8]. The existence of a time evolution and equilibrium states for various anharmonic crystal systems will also be discussed [9].Supported in part by AFOSR Grant No. 73-2430B.