Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling
KAUST Grant NumberKUK-I1-007-43
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AbstractConservation equations governed by a nonlocal interaction potential generate aggregates from an initial uniform distribution of particles. We address the evolution and formation of these aggregating 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 complementary solution methods, a continuous pseudoinverse method and a discrete stochastic lattice approach, and formally show a connection between the two. Interesting aggregation patterns involving 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 is observed in the evolution to steady state in both the continuous and discrete approaches. The discrete approach is found to be remarkably robust to modifications in movement rules, related to the potential function. The comparable evolution dynamics and steady states of the discrete model with the continuum model suggest that the discrete stochastic approach is a promising way of probing aggregation patterns arising from two- and three-dimensional nonlocal interaction conservation equations. © 2012 American Physical Society.
CitationHackett-Jones EJ, Landman KA, Fellner K (2012) Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling. Phys Rev E 85. Available: http://dx.doi.org/10.1103/PhysRevE.85.041912.
SponsorsThis work was supported by the Australian Research Council Discovery Grant (Kerry Landman). K.L. acknowledges support by an ARC Fellowship. Klemens Fellner was supported by Award No. KUK-I1-007-43 of Peter A. Markowich, University of Cambridge, made by King Abdullah University of Science and Technology (KAUST). We thank Barry Hughes for many useful discussions on this work and related matters. We also thank Federico Frascoli for his assistance.
PublisherAmerican Physical Society (APS)
JournalPhysical Review E
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