Microscopic derivation of the Keller-Segel equation in the sub-critical regime

TL;DR

This paper rigorously derives the two-dimensional Keller-Segel equation from a stochastic particle system in the sub-critical regime, providing quantitative bounds on the approximation accuracy.

Contribution

It introduces a regularized Coulomb interaction with a cutoff depending on N and establishes a quantitative link between particle trajectories and the mean-field limit.

Findings

Quantitative bounds on particle and mean-field trajectory differences

Derivation valid for sub-critical chemosensitivity $ ext{chi} < 8 \pi$

Regularization of Coulomb force with cutoff $N^{- ext{alpha}}$ for $ ext{alpha} ext{ in } (0, 1/2)$

Abstract

We derive the two-dimensional Keller-Segel equation from a stochastic system of $N$ interacting particles in the case of sub-critical chemosensitivity $\chi < 8 \pi$. The Coulomb interaction force is regularised with a cutoff of size $N^{- \alpha}$, with arbitrary $\alpha \in (0, 1 / 2)$. In particular we obtain a quantitative result for the maximal distance between the real and mean-field $N$-particle trajectories.