Glauber Dynamics in Continuum: A Constructive Approach to Evolution of States

TL;DR

This paper develops a constructive framework for describing the evolution of states in continuum Glauber dynamics, using both microscopic and mesoscopic approaches, including correlation functions and Vlasov scaling.

Contribution

It introduces a novel method combining linear equations for correlation functions with Vlasov scaling to analyze infinite particle systems in continuum.

Findings

Solution for initial correlation functions of Poisson measures is given by a nonlinear differential equation.

Evolution described on both micro- and mesoscopic levels using Banach space techniques.

The approach provides a constructive method for analyzing continuum Glauber dynamics.

Abstract

The evolutions of states is described corresponding to the Glauber dynamics of an infinite system of interacting particles in continuum. The description is conducted on both micro- and mesoscopic levels. The microscopic description is based on solving linear equations for correlation functions by means of an Ovsjannikov-type technique, which yields the evolution in a scale of Banach spaces. The mesoscopic description is performed by means of the Vlasov scaling, which yields a linear infinite chain of equations obtained from those for the correlation function. Its main peculiarity is that, for the initial correlation function of the inhomogeneous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type.