THE VLASOV CONTINUUM LIMIT FOR THE CLASSICAL MICROCANONICAL ENSEMBLE

Abstract

For classical Hamiltonian N-body systems with mildly regular pair interaction potential (in particular, [Formula: see text] integrability is required), it is shown that when N → ∞ in a fixed bounded domain Λ ⊂ ℝ3, with energy [Formula: see text] scaling as [Formula: see text], then Boltzmann's ergodic ensemble entropy [Formula: see text] has the asymptotic expansion SΛ(N,N2ε) = - N ln N + sΛ(ε) N + o(N). Here, the N ln N term is combinatorial in origin and independent of the rescaled Hamiltonian, while sΛ(ε) is the system-specific Boltzmann entropy per particle, i.e. –sΛ(ε) is the minimum of Boltzmann's H function for a perfect gas of energy ε 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 [Formula: see text] is studied rather than its inverse [Formula: see text]; (b) no regularization of the microcanonical measure [Formula: see text] is invoked, and (c) no detour via the canonical ensemble. The proofs hold irrespective of whether microcanonical and canonical ensembles are equivalent or not.