Discrete-to-continuum limits of particles with an annihilation rule

TL;DR

This paper rigorously establishes a discrete-to-continuum limit for particle systems with annihilation, enabling analysis of phenomena like vortices and dislocations where particles of opposite sign collide and vanish.

Contribution

It introduces the first rigorous framework for discrete-to-continuum limits that incorporate particle annihilation, bridging a gap in the mathematical understanding of such systems.

Findings

Empirical measures of discrete dynamics satisfy the continuum evolution equation with implicit annihilation.

A mild initial separation condition ensures the limiting density solves the continuum equation.

The approach applies to systems with singular same-sign interactions and regular opposite-sign interactions.

Abstract

In the recent trend of extending discrete-to-continuum limit passages for gradient flows of single-species particle systems with singular and nonlocal interactions to particles of opposite sign, any annihilation effect of particles with opposite sign has been side-stepped. We present the first rigorous discrete-to-continuum limit passage which includes annihilation. This result paves the way to applications such as vortices, charged particles, and dislocations. In more detail, the discrete setting of our discrete-to-continuum limit passage is given by particles on the real line. Particles of the same type interact by a singular interaction kernel; those of opposite sign interact by a regular one. If two particles of opposite sign collide, they annihilate, i.e., they are taken out of the system. The challenge for proving a discrete-to-continuum limit is that annihilation is an intrinsically discrete effect where particles vanish instantaneously in time, while on the continuum scale the mass of the particle density decays continuously in time. The proof contains two novelties: (i) the observation that empirical measures of the discrete dynamics (with annihilation rule) satisfy the continuum evolution equation that only implicitly encodes annihilation, and (ii) the fact that, by imposing a relatively mild separation assumption on the initial data, we can identify the limiting particle density as a solution to the same continuum evolution equation.