The Microscopic Foundations of Vlasov Theory for Jellium-Like Newtonian N-Body Systems

摘要

The kinetic equations of Vlasov theory, in the weak formulation, are rigorously shown to govern the N → ∞limit of the Newtonian dynamics of D ≥ 2-dimensional N-body systems with attractive harmonic pair interactions and locally integrable repulsive inverse power law pair interactions, provided a mild higher moment hypothesis on the forces (which is shown to propagate globally in time for each N) will hold uniformly in N at later times if it holds uniformly in N initially (the uniformity in N of this moment condition is demonstrated to hold for an open set of initial data). Logarithmic interactions are included as a limiting case. The proof is based on the Liouville equation, more precisely the first member of the pertinent BBGKY hierarchy, and does not invoke the Hewitt-Savage theorem, nor any regularization of the interactions. In addition, a rigorous proof of the virial theorem and of some of its interesting ramifications is given.