Statistical equilibrium dynamics

We study the mean-field thermodynamic limit for a class of isolated Newtonian N-body systems whose Hamiltonian admits several additional integrals of motion. Examples are systems which are isomorphic to plasma models consisting of one specie of charged particles moving in a neutralizing uniform background charge. We find that in the limit of infinitely many particles the stationary ensemble measures with prescribed values of the integrals of motion are supported on the set of maximum entropy solutions of a (time-independent) nonlinear fixed point equation of mean-field type. Each maximum entropy solution of this fixed point equation can in turn be either a static or a stationary solution for the entropy-con serving Vlasov evolution, or even belong to a one-dimensional orbit of maximum entropy solutions which evolve into one another by the Vlasov dynamics. In short, the macrostates of individual members of an equilibrium ensemble are not necessarily themselves in a state of global statistical equilibrium in the strict sense. Yet they are always locally in thermodynamic equilibrium, and always global maximizers of the pertinent maximum entropy principle.