Discrete-to-continuum convergence of charged particles in 1D with annihilation

Abstract

We consider a system of charged particles moving on the real line driven by electrostatic interactions. Since we consider charges of both signs, collisions might occur in finite time. Upon collision, some of the colliding particles are effectively removed from the system (annihilation). The two applications we have in mind are vortices and dislocations in metals. In this paper we reach two goals. First, we develop a rigorous solution concept for the interacting particle system with annihilation. The main innovation here is to provide a careful management of the annihilation of groups of more than two particles, and we show that the definition is consistent by proving existence, uniqueness, and continuous dependence on initial data. The proof relies on a detailed analysis of ODE trajectories close to collision, and a reparametrization of vectors in terms of the moments of their elements. Secondly, we pass to the many-particle limit (discrete-to-continuum), and recover the expected limiting equation for the particle density. Due to the singular interactions and the annihilation rule, standard proof techniques of discrete-to-continuum limits do not apply. In particular, the framework of measures seems unfit. Instead, we use the one-dimensional feature that both the particle system and the limiting PDE can be characterized in terms of Hamilton--Jacobi equations. While our proof follows a standard limit procedure for such equations, the novelty with respect to existing results lies in allowing for stronger singularities in the particle system by exploiting the freedom of choice in the definition of viscosity solutions.