Convergence and non-convergence of many-particle evolutions with multiple signs

TL;DR

This paper investigates the convergence of many-particle systems with positive and negative particles interacting via a singular potential, showing how regularisation scale affects the derivation of continuum PDE models like the Groma-Balogh equations.

Contribution

The paper establishes conditions under which the empirical measures of interacting particles converge to continuum PDEs and provides counterexamples for different regularisation scales.

Findings

Empirical measures converge to Groma-Balogh PDEs when regularisation scale decreases slowly.

Fast decay of regularisation scale leads to divergence from the PDEs.

The validity of continuum models depends critically on the regularisation scale.

Abstract

We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale $\delta_n>0$, which converges to zero as the number of particles $n$ tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma-Balogh equations. In the proof we rely on the theory of $\lambda$-convex gradient flows, a priori estimates for the Groma-Balogh equations and Orlicz spaces. The proof require $\delta_n$ to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if $\delta_n$ converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma-Balogh equations. These results show how the validity of the Groma-Balogh equations as the limit of many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.