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New Perspectives in the Study of Swarming Systems

New Perspectives in the Study of Swarming Systems. Cristi á n Huepe Unaffiliated NSF Grantee - Chicago, IL. USA. Collaborators: Maximino Aldana, Paul Umbanhowar, Hernan Larralde, V. M. Kenkre, V. Dossetti. This work was supported by the National Science Foundation under Grant No. DMS-0507745.

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New Perspectives in the Study of Swarming Systems

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  1. New Perspectives in the Study of Swarming Systems Cristián Huepe Unaffiliated NSF Grantee - Chicago, IL. USA. Collaborators: Maximino Aldana, Paul Umbanhowar, Hernan Larralde,V. M. Kenkre, V. Dossetti. This work was supported by the National Science Foundation under Grant No. DMS-0507745.

  2. Talk Outline • Overview of Swarming Systems Research • Biological and technological motivation • Various theoretical approaches • Agent-Based Modeling • Minimal agent-based models • Order parameters and phase transition • Intermittency and Clustering • Experimental and numerical results • The two-particles case • The N-particle case • The Network Approach • Motivation: “small-world” effect • Analytic solution • Future Challenges and Experiments

  3. Biological & technological motivation Biological agents From Iain Couzin’s group: http://www.princeton.edu/~icouzin Decentralized robots From James McLurkin’s group: http://people.csail.mit.edu/jamesm/swarm.php

  4. Various approaches • Biology:Iain Couzin (Oxford/Princeton), Stephen Simpson (U of Sidney), Julia Parrish, Daniel Grünbaum (U of Washington), Steven Viscido (U of South Carolina), Leah Edelstein-Keshet (U of British Columbia), Charlotte Hemelrijk (U of Groningen) • Engineering:Naomi Leonard (Princeton), Richard Murray (CALTECH), Reza Olfati-Saber (Dartmouth College), Ali Jadbabaie (U of Pennsylvania), Stephen Morse (Yale U), Kevin Lynch, Randy Freeman (Northwestern U), Francesco Bullo (UCSB), Vijay Kumar (U of Pennsylvania) • Applied math / Non-equilibrium Physics: Chad Topaz, Andrea Bertozzi, Maria D’Orsogna (UCLA), Herbert Levine (UCSD), Tamás Vicsek (Eötvös Loránd U), Hugues Chaté (CEA-Saclay), Maximino Aldana (UNAM), Udo Erdmann (Helmholtz Association), Bruno Eckhardt (Philipps-U Marburg), Edward Ott (U of Maryland)

  5. Minimal agent-based models (10) • Vicsek et al. noise • Original Vicsek Algorithm (OVA) • Standard Vicsek Algorithm (SVA) • Guillaume-Chaté Algorithm (GCA)

  6. Order parameters & phase transition • Degree of alignment(magnetization): • Local density: • Distance to nearest neighbor:

  7. Degree of alignment vs. amount of noise • Local density vs. amount of noise

  8. (Grégoire & Chaté: PRL 90(2)025702) • GCA: 1st order phase transition? • Observations: • Apparent 2nd order phase transition for large N • SVA appears to have larger finite-size effect • GCA appears to present similar transition • SVA and GCA: Unrealistic local densities

  9. Intermittency and Clustering • Experiments • Simulations

  10. The two-particle case • 1st passage problem in a 1D random walk. • We compute the continuous approximation • Diffusion equation with • Analytic solution in Laplace space for: • Distribution of laminar intervals

  11. The N-particle case • Alignment vs. time • N=5000 agents • N=500 agents • N=2 agents • Probability distribution of the degree of alignment

  12. Clustering Analysis • Power-law cluster size (agent number) distribution • No characteristic cluster size • Power-law cluster size transition prob. • Of belonging to cluster of size ‘n’ at ‘t’ and ‘n+n’ at ‘t+1’

  13. The Network Approach • Motivation: We replace • Moving agents by fixed nodes. • Effective long-range interactions by a few long-range connections. • Each node linked with probability 1-p to one of its K neighbors and p to any other node. • Small-world effect: • 1% of long range connections • Phase with long-range order appears p = 0.1

  14. Analytic Solution • Mean-field approximation • Vicsek time-step and order parameter: • Order parameter: • The calculation requires: • Expressing PDFs in terms of moments • A random-walk analogy • Central limit theorem • Expansion about the phase transition point

  15. Results • SVA: 2nd order phase transition with critical behavior: • GCA: 1st order phase transition Vicsek AlgorithmGuillaume-Chate Algorithm

  16. Future Challenges & Experiments • Examine a more rigorous connection between the network model and the self-propelled system • Understand the effects of intermittency in the swarm’s non-equilibrium dynamics • Consider new order parameters • New quantitative experiments • (With Paul Umbanhowar)

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