Kawasaki dynamics in continuum: micro- and mesoscopic descriptions

TL;DR

This paper develops a rigorous mathematical framework for describing the dynamics of an infinite system of interacting particles in continuum space at micro- and mesoscopic levels, including correlation functions and kinetic equations.

Contribution

It introduces a novel approach to derive and analyze correlation functions and kinetic equations for particle systems, bridging micro- and mesoscopic descriptions.

Findings

Existence and uniqueness of evolution of states in sub-Poissonian measures.

Derivation of a Vlasov-type kinetic equation from correlation functions.

Proven global existence of solutions to the kinetic equation.

Abstract

The dynamics of an infinite system of point particles in $\mathbb{R}^d$, 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 $\mu_0 \mapsto \mu_t$ 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 $k_0 \mapsto k_t$, $t\in [0,T)$, in a scale of Banach spaces; (b) proving that each $k_t$ is a correlation function for a unique measure $\mu_t$. 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 $\varrho_t$, $t\in [0,+\infty)$.