THE VLASOV CONTINUUM LIMIT FOR THE CLASSICAL MICROCANONICAL ENSEMBLE

For classical Hamiltonian N-body systems with mildly regular pair interaction potential (in particular, L-loc(2) integrability is required), it is shown that when N -> infinity in a fixed bounded domain Lambda subset of R-3, with energy epsilon scaling as epsilon proportional to N-2, then Boltzmann's ergodic ensemble entropy S-Lambda(N, epsilon) has the asymptotic expansion S-Lambda(N, N-2 epsilon) = -N ln N + s(Lambda)(epsilon)N + o(N). Here, the N ln N term is combinatorial in origin and independent of the rescaled Hamiltonian, while s(Lambda)(epsilon) is the system-specific Boltzmann entropy per particle, i.e. - s(Lambda)(epsilon) is the minimum of Boltzmann's H function for a perfect gas of energy e subjected to a combination of externally and self-generated fields. It is also shown that any limit point of the n-point marginal ensemble measures is a linear convex superposition of n-fold products of the H-function-minimizing one-point functions. The proofs are direct, in the sense that (a) the map epsilon bar right arrow S(epsilon) is studied rather than its inverse S bar right arrow E(S); (b) no regularization of the microcanonical measure delta(epsilon - H) is invoked, and (c) no detour via the canonical ensemble. The proofs hold irrespective of whether microcanonical and canonical ensembles are equivalent or not.