Global existence and mean-field limit for a stochastic interacting particle system of signed Coulomb charges

TL;DR

This paper investigates a stochastic particle system with positive and negative charges interacting via Coulomb potential, establishing well-posedness, collision behavior, and mean-field limits, revealing features of both repulsive and attractive cases.

Contribution

It introduces a novel analysis of signed Coulomb particle systems, extending understanding beyond identical particle interactions and employing a new proof approach inspired by prior work.

Findings

Well-posedness of the signed Coulomb system

Characterization of collision types

Mean-field limit as particle number grows

Abstract

We study a system of stochastic differential equations with singular drift which describes the dynamics of signed particles in two dimensions interacting by the Coulomb potential. In contrast to the well-studied cases of identical particles that either all repel each other or all attract each other, this system contains both `positive' and `negative' particles. Equal signs repel and opposite signs attract each other; apart from the sign, the potential is the same. We derive results on well-posedness of the system, on the type of collisions that can occur, and on the mean-field limit as the number of particles tends to infinity. Our results demonstrate that the signed system shares features of both the fully repulsive and the fully attractive cases. Our proof method is inspired by the work of Fournier and Jourdain (The Annals of Applied Probability, 27, pp. 2807-2861, 2017) on the fully attractive case; we construct an approximate system of equations, establish uniform estimates, and use tightness to pass to limits.