The Vlasov continuum limit for the classical microcanonical ensemble

TL;DR

This paper rigorously derives the asymptotic behavior of the microcanonical entropy for large classical Hamiltonian systems, revealing a universal leading term and characterizing the structure of limit measures without relying on ensemble equivalence.

Contribution

It provides a direct, regularization-free proof of the microcanonical entropy asymptotics and describes the structure of limit measures as superpositions of product measures.

Findings

Asymptotic expansion of entropy: -N log N + s(e) N + o(N)

Limit measures are convex superpositions of product measures minimizing the H-function

Proofs are direct, avoiding regularization and ensemble transformations

Abstract

For classical Hamiltonian N-body systems with mildly regular pair interaction potential it is shown that when N tends to infinity in a fixed bounded domain, with energy E scaling quadratically in N proportional to e, then Boltzmann's ergodic ensemble entropy S(N,E) has the asymptotic expansion S(N,E) = - N log N + s(e) N + o(N); here, the N log N term is combinatorial in origin and independent of the rescaled Hamiltonian while s(e) is the system-specific Boltzmann entropy per particle, i.e. -s(e) 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 E to S(E) is studied rather than its inverse S to E(S); (b) no regularization of the microcanonical measure Dirac(E-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.