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

The kinetic equations of Vlasov theory, in the weak formulation, are rigorously shown to govern the $N\rightarrow \infty $ limit of the Newtonian dynamics of $D\ge 2$ -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.