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

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.

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