Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling.

Institute for Mathematics and Scientific Computing,University of Graz, A-8010 Graz, Heinrichstr. 36, Austria(Dated: October 18, 2011)Conservation equations governed by a non-local interaction potential generate aggregates froman initial uniform distribution of particles. We address the evolution and formation of these aggre-gating steady states when the interaction potential has both attractive and repulsive singularities.Currently, no existence theory for such potentials is available. We develop and compare two com-plementary solution methods, a continuous pseudo-inverse method and a discrete stochastic latticeapproach, and formally show a connection between the two. Interesting aggregation patterns in-volving multiple peaks for a simple doubly-singular attractive-repulsive potential are determined.For a swarming Morse potential, characteristic slow-fast dynamics in the scaled inverse energy isobserved in the evolution to steady state in both the continuous and discrete approaches. Thediscrete approach is found to be remarkably robust to modifications in movement rules, related tothe potential function. The comparable evolution dynamics and steady states of the discrete modelwith the continuum model suggest that the discrete stochastic approach is a promising way of prob-ing aggregation patterns arising from two and three-dimensional non-local interaction conservationequations.