On the stability of equilibrium states of finite classical systems

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(d x) = f(H)d x;d x 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.