Communications in Mathematical Sciences

Volume 16 (2018)

Number 2

Coalescing particle systems and applications to nonlinear Fokker–Planck equations

Pages: 463 – 490



Gleb Zhelezov (Department of Mathematics, University of Arizona, Tucson, Az., U.S.A.)

Ibrahim Fatkullin (Department of Mathematics, University of Arizona, Tucson, Az., U.S.A.)


We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker–Planck equation, such as the Patlak–Keller–Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller–Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.


coalescing particles, coarsening, Bessel process, Keller–Segel, multi-component Keller–Segel, Fokker–Planck, grid-particle method, blow-up, chemotaxis, Vlasov–Poisson

2010 Mathematics Subject Classification

35K58, 35Q83, 35Q92, 45G05, 60H30, 60H35, 65C35, 82C21, 82C22, 82C31, 82C80, 92C17

Received 19 April 2017

Accepted 1 December 2017

Published 14 May 2018