2. 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.

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

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

5. 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)

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

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

8. Degree of alignment vs. amount of noise Local density vs. amount of noise

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

10. Intermittency and Clustering ExperimentsSimulations

11. The two-particle case 1 st 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 Distribution of laminar intervals

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

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

14. The Network Approach Motivation: We replace Moving agents by fixed nodes. Moving agents by fixed nodes. Effective long-range interactions by a few long-range connections. 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 1% of long range connections Phase with long-range order appears Phase with long-range order appears p = 0.1

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

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

17. 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) (With Paul Umbanhowar)