  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.
