Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling

Handle URI:
http://hdl.handle.net/10754/597477
Title:
Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling
Authors:
Hackett-Jones, Emily J.; Landman, Kerry A.; Fellner, Klemens
Abstract:
Conservation 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.
Citation:
Hackett-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.
Publisher:
American Physical Society (APS)
Journal:
Physical Review E
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
17-Apr-2012
DOI:
10.1103/PhysRevE.85.041912
PubMed ID:
22680503
Type:
Article
ISSN:
1539-3755; 1550-2376
Sponsors:
This 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.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorHackett-Jones, Emily J.en
dc.contributor.authorLandman, Kerry A.en
dc.contributor.authorFellner, Klemensen
dc.date.accessioned2016-02-25T12:40:30Zen
dc.date.available2016-02-25T12:40:30Zen
dc.date.issued2012-04-17en
dc.identifier.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.en
dc.identifier.issn1539-3755en
dc.identifier.issn1550-2376en
dc.identifier.pmid22680503en
dc.identifier.doi10.1103/PhysRevE.85.041912en
dc.identifier.urihttp://hdl.handle.net/10754/597477en
dc.description.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.en
dc.description.sponsorshipThis 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.en
dc.publisherAmerican Physical Society (APS)en
dc.titleAggregation patterns from nonlocal interactions: Discrete stochastic and continuum modelingen
dc.typeArticleen
dc.identifier.journalPhysical Review Een
dc.contributor.institutionUniversity of Melbourne, Parkville, Australiaen
dc.contributor.institutionKarl-Franzens-Universitat Graz, Graz, Austriaen
kaust.grant.numberKUK-I1-007-43en

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