  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.
