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