Collisions of the supercritical Keller–Segel particle system

Abstract

We study a particle system naturally associated to the $2$-dimensional Keller–Segel equation. It consists of $N$ Brownian particles in the plane, interacting through a binary attraction in $\theta /(Nr)$, where $r$ stands for the distance between two particles. When the intensity $\theta$ of this attraction is greater than $2$, this particle system explodes in finite time. We assume that $N>3\theta$ and study in detail what happens near explosion. There are two slightly different scenarios, depending on the values of $N$ and $\theta$, here is one: at explosion, a cluster consisting of precisely $k_0$ particles emerges, for some deterministic $k_0\ge 7$ depending on $N$ and $\theta$. Just before explosion, there are infinitely many $(k_0-1)$-ary collisions. There are also infinitely many $(k_0-2)$-ary collisions before each $(k_0-1)$-ary collision. And there are infinitely many binary collisions before each $(k_0-2)$-ary collision. Finally, collisions of subsets of $3,\dots ,k_0-3$ particles never occur. The other scenario is similar except that there are no $(k_0-2)$-ary collisions.