Kawasaki Dynamics in Continuum: Micro- and Mesoscopic Descriptions

Abstract

The dynamics of an infinite system of point particles in ℝ[superscript d], which hop and interact with each other, is described at both micro- and mesoscopic levels. The states of the system are probability measures on the space of configurations of particles. For a bounded time interval [0,T), the evolution of states μ[subscript 0]↦μ[subscript t] is shown to hold in a space of sub-Poissonian measures. This result is obtained by: (a) solving equations for correlation functions, which yields the evolution k[subscript 0]↦k[subscript t], t∈[0,T), in a scale of Banach spaces; (b) proving that each k[subscript t] is a correlation function for a unique measure μ[subscript t]. The mesoscopic theory is based on a Vlasov-type scaling, that yields a mean-field-like approximate description in terms of the particles’ density which obeys a kinetic equation. The latter equation is rigorously derived from that for the correlation functions by the scaling procedure. We prove that the kinetic equation has a unique solution ϱ[subscript t], t∈[0,+∞).