Many-particle limit for a system of interaction equations driven by Newtonian potentials

TL;DR

This paper studies a two-species particle system with nonlocal Newtonian interactions, analyzing collision behavior and proving convergence of the empirical measure to a PDE-based gradient flow solution in the many-particle limit.

Contribution

It establishes existence, collision analysis, and convergence of the particle system to a PDE model with nonlocal interactions, extending understanding of multi-species dynamics.

Findings

Empirical measures converge to a PDE gradient flow solution.

Collision behavior is characterized within the particle system.

Uniform estimates of density norms are obtained for convergence analysis.

Abstract

We consider a discrete particle system of two species coupled through nonlocal interactions driven by the one-dimensional Newtonian potential, with repulsive self-interaction and attractive cross-interaction. After providing a suitable existence theory in a finite-dimensional framework, we explore the behaviour of the particle system in case of collisions and analyse the behaviour of the solutions with initial data featuring particle clusters. Subsequently, we prove that the empirical measure associated to the particle system converges to the unique 2-Wasserstein gradient flow solution of a system of two partial differential equations (PDEs) with nonlocal interaction terms in a proper measure sense. The latter result uses uniform estimates of the $L^m$-norms of a piecewise constant reconstruction of the density using the particle trajectories.