Non-Equilibrium Dynamics of Dyson's Model with an Infinite Number of Particles

TL;DR

This paper establishes conditions under which Dyson's model with infinitely many particles is well-defined and explores its non-equilibrium dynamics, including relaxation to a stationary determinantal point process with the sine kernel.

Contribution

It provides sufficient conditions for the well-definedness of Dyson's model with infinite particles and analyzes its non-equilibrium relaxation dynamics.

Findings

Conditions for initial configurations ensuring well-defined infinite Dyson's model

Demonstration of relaxation to the sine kernel determinantal point process

Extension of the model to non-equilibrium initial states

Abstract

Dyson's model is a one-dimensional system of Brownian motions with long-range repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant $\beta/2$. We give sufficient conditions for initial configurations so that Dyson's model with $\beta=2$ and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of $\Z$ is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.