On the stability of equilibrium states of finite classical systems

Abstract

The state of a system is characterized, in statistical mechanics, by a measure ω on Γ, the phase space of the system (i.e., by an ensemble). To represent an equilibrium state, the measure must be stationary under the time evolution induced by the systems Hamiltonian H (x), x‐Γ. An example of such a measure is ω (dx) = f (H) dx;dx is the Liouville (Lebesgue) measure and f (H (x)) is the ensemble density. For ’’nonergodic’’ systems there are also other stationary measures with ensemble densities, e.g., for integrable dynamical systems the density can be a function of any of the constants of the motion. We show, however, that the requirement that the equilibrium measure have a certain type of ’’stability’’ singles out, in the typical case, densities which depend only on H.