Quantum Separation of Variables and Multi-component Dyson Brownian Motion

TL;DR

This paper introduces a generalized Coulomb gas model with multiple charge species undergoing Brownian motion, extending Dyson's model to describe non-equilibrium dynamics of charges with specific interaction conditions.

Contribution

It develops a new Coulomb gas model with multiple charge types and provides an exact mathematical description of their Brownian dynamics under electrostatic interactions.

Findings

Defines a Coulomb gas with arbitrary charge species and Brownian motion.

Provides an exact mathematical framework for multi-charge Dyson Brownian motion.

Generalizes Dyson's model to include different charge magnitudes and interactions.

Abstract

A new type of Coulomb gas is defined, consisting of arbitrary numbers of point charges of two species executing Brownian motions under the influence of their mutual electrostatic repulsion. Being a generalization of a model of identical particles introduced by Dyson as a dynamical system describing non-equilibrium state of various random matrix ensembles, our system gives an exact mathematical description of the Brownian motion of charges of magnitudes $ Q_1 $ and $ Q_2 $, such that $ \beta Q_1 Q_2 = 1 $ on the line or circle, where $\beta$ is an inverse temperature of the gas.