Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit

TL;DR

This paper studies a non-equilibrium stochastic dynamics of binary particle jumps in continuum space, proving the existence of correlation function evolution and deriving a Vlasov-type scaling limit leading to a generalized Boltzmann equation.

Contribution

It introduces a non-equilibrium binary jump process in continuum and establishes a Vlasov-type scaling limit resulting in a nonlinear Boltzmann equation.

Findings

Existence of correlation function evolution over finite time.

Derivation of a Vlasov-type mesoscopic limit.

Connection to a generalized Boltzmann equation.

Abstract

Let $\Gamma$ denote the space of all locally finite subsets (configurations) in $\mathbb R^d$. A stochastic dynamics of binary jumps in continuum is a Markov process on $\Gamma$ in which pairs of particles simultaneously hop over $\mathbb R^d$. We discuss a non-equilibrium dynamics of binary jumps. We prove the existence of an evolution of correlation functions on a finite time interval. We also show that a Vlasov-type mesoscopic scaling for such a dynamics leads to a generalized Boltzmann non-linear equation for the particle density.