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

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 β/2. We give sufficient conditions for initial configurations so that Dyson’s model with β = 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 $${\mathbb{Z}}$$ is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.