The Vlasov continuum limit for the classical microcanonical ensemble

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.

Final version; a few typos corrected; minor changes in the presentation

References in corpus (6)

• The large deviation approach to statistical mechanics
• Phase transitions in self-gravitating systems. Self-gravitating fermions and hard spheres models
• The generalized canonical ensemble and its universal equivalence with the microcanonical ensemble
• Dynamics and thermodynamics of systems with long-range interactions: interpretation of the different functionals
• Relaxation to a Perpetually Pulsating Equilibrium
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• Monotonicity of quantum ground state energies: Bosonic atoms and stars
• Classical Density Functional Theory: Representability and Universal Bounds