New Perspectives in the New Perspectives in the Study of Swarming SystemsStudy of Swarming Systems

CristiCristiáán Huepen Huepe Unaffiliated NSF Grantee - Chicago, IL. USA.Unaffiliated NSF Grantee - Chicago, IL. USA.

Collaborators: Maximino Aldana, Collaborators: Maximino Aldana, Paul Paul Umbanhowar, Hernan Larralde,Umbanhowar, Hernan Larralde, V. M. Kenkre, V. Dossetti.V. M. Kenkre, V. Dossetti.

This work was supported by the National Science Foundation under Grant No. DMS-0507745.

Talk OutlineTalk OutlineOverview of Swarming Systems ResearchOverview of Swarming Systems Research

Biological and technological motivationBiological and technological motivation Various theoretical approachesVarious theoretical approaches

Agent-Based ModelingAgent-Based Modeling Minimal agent-based modelsMinimal agent-based models Order parameters and phase transitionOrder parameters and phase transition

Intermittency and ClusteringIntermittency and Clustering Experimental and numerical resultsExperimental and numerical results The two-particles caseThe two-particles case The N-particle caseThe N-particle case

The Network ApproachThe Network Approach Motivation: “small-world” effectMotivation: “small-world” effect Analytic solutionAnalytic solution

Future Challenges and ExperimentsFuture Challenges and Experiments

Biological & technological motivationBiological & 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

Bio

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Various approachesVarious approaches

Biology:Biology: Iain CouzinIain Couzin (Oxford/Princeton), (Oxford/Princeton), Stephen SimpsonStephen Simpson (U of (U of Sidney), Sidney), Julia ParrishJulia Parrish, , Daniel GrDaniel Grünbaumünbaum (U of Washington), (U of Washington), Steven Steven Viscido (U of South Carolina), Viscido (U of South Carolina), Leah Edelstein-KeshetLeah Edelstein-Keshet (U of British (U of British Columbia), Columbia), Charlotte HemelrijkCharlotte Hemelrijk (U of Groningen) (U of Groningen)

Engineering:Engineering: Naomi LeonardNaomi Leonard (Princeton), (Princeton), Richard MurrayRichard Murray (CALTECH), (CALTECH), Reza Olfati-SaberReza Olfati-Saber (Dartmouth College), (Dartmouth College), Ali JadbabaieAli Jadbabaie (U of Pennsylvania), (U of Pennsylvania), Stephen MorseStephen Morse (Yale U), (Yale U), Kevin LynchKevin Lynch, , Randy Randy FreemanFreeman (Northwestern U), (Northwestern U), Francesco BulloFrancesco Bullo (UCSB), Vijay Kumar (UCSB), Vijay Kumar (U of Pennsylvania)(U of Pennsylvania)

Applied math / Non-equilibrium Physics: Applied math / Non-equilibrium Physics: Chad TopazChad Topaz, , Andrea Andrea Bertozzi, Maria D’OrsognaBertozzi, Maria D’Orsogna (UCLA), (UCLA), Herbert LevineHerbert Levine (UCSD), (UCSD), TamTamáás s VicsekVicsek (E (Eötvös Loránd U), ötvös Loránd U), Hugues ChatéHugues Chaté (CEA-Saclay), (CEA-Saclay), Maximino Maximino AldanaAldana (UNAM), (UNAM), Udo ErdmannUdo Erdmann (Helmholtz Association), (Helmholtz Association), Bruno Bruno EckhardtEckhardt (Philipps-U Marburg), (Philipps-U Marburg), Edward OttEdward Ott (U of Maryland) (U of Maryland)

Minimal agent-based modelsMinimal agent-based modelsVicsek Vicsek et al.et al. noise noise

Original Vicsek Algorithm (Original Vicsek Algorithm (OVAOVA))

Standard Vicsek Algorithm (Standard Vicsek Algorithm (SVASVA))

Guillaume-ChatGuillaume-Chaté Algorithm (é Algorithm (GCAGCA))

(10)

Order parameters & phase transitionOrder parameters & phase transition

Degree of alignmentDegree of alignment (magnetization)(magnetization)::

Local density:Local density:

Distance to nearest neighbor:Distance to nearest neighbor:

.8.0,1.0

,4.0,1000

1:Parameters

0

v

N

r

Degree of alignment vs. amount of noiseDegree of alignment vs. amount of noise

Local density vs. amount of noiseLocal density vs. amount of noise

GCA: 1GCA: 1stst order phase transition? order phase transition?

Observations:Observations: Apparent 2Apparent 2ndnd order phase transition for large N order phase transition for large N SVA appears to have larger finite-size effectSVA appears to have larger finite-size effect GCA appears to present similar transitionGCA appears to present similar transition SVA and GCA: Unrealistic local densitiesSVA and GCA: Unrealistic local densities

(Grégoire & Chaté: PRL 90(2)025702)

Intermittency and ClusteringIntermittency and ClusteringExperimentsExperiments

SimulationsSimulations

.0.1,1.0,4.0

,1000,1:Parameters

0 v

Nr

The two-particle caseThe two-particle case11stst passage problem in a 1D random walk. passage problem in a 1D random walk.We compute the continuous approximationWe compute the continuous approximationDiffusion equation withDiffusion equation with

Analytic solution in Laplace space for:Analytic solution in Laplace space for: Distribution of laminar intervalsDistribution of laminar intervals

rx

xx

2

2 ,,

x

txcD

t

txc

tD 22

The N-particle caseThe N-particle case

Alignment vs. timeAlignment vs. time N=5000 agentsN=5000 agents

N=500 agentsN=500 agents N=2 agentsN=2 agents

Probability distribution Probability distribution of the degree of of the degree of alignmentalignment

Clustering AnalysisClustering AnalysisPower-law cluster size Power-law cluster size (agent number) distribution(agent number) distribution

No characteristic cluster sizeNo characteristic cluster size

Power-law cluster size Power-law cluster size transition prob.transition prob.

Of belonging to Of belonging to cluster of size ‘n’ at cluster of size ‘n’ at ‘‘t’ and ‘n+n’ at ‘t+1’t’ and ‘n+n’ at ‘t+1’

The Network ApproachThe Network ApproachMotivation: We replaceMotivation: We replace

Moving agents by fixed nodes.Moving agents by fixed nodes. EffectiveEffective long-range interactions by a few long-range connections. long-range interactions by a few long-range connections.

Each node linked with probability Each node linked with probability 1-p1-p to one of its K neighbors and to one of its K neighbors and pp to any other node.to any other node.

Small-world effect:Small-world effect: 1% of long range connections1% of long range connections Phase with long-range order appearsPhase with long-range order appears

p = 0.1

Mean-field approximationMean-field approximation Vicsek time-step and order parameter:Vicsek time-step and order parameter:

Order parameter:Order parameter:

The calculation requires:The calculation requires: Expressing PDFs in terms Expressing PDFs in terms

of momentsof moments A random-walk analogyA random-walk analogy Central limit theoremCentral limit theorem Expansion about theExpansion about the

phase transition pointphase transition point

22

2 sincos

dPdP

2

2

;;1;

dtPPtPtP KK

Analytic SolutionAnalytic Solution

ResultsResults

SVASVA: 2: 2ndnd order phase transition with critical behavior: order phase transition with critical behavior:

GCA:GCA: 1 1stst order phase transition order phase transition

Vicsek AlgorithmVicsek Algorithm Guillaume-Chate AlgorithmGuillaume-Chate Algorithm

2sin c

c K

Future Challenges & ExperimentsFuture Challenges & Experiments

Examine a more rigorous connection between the Examine a more rigorous connection between the network model and the self-propelled systemnetwork model and the self-propelled system

Understand the effects of intermittency in the swarm’s Understand the effects of intermittency in the swarm’s non-equilibrium dynamicsnon-equilibrium dynamics

Consider new order parametersConsider new order parameters

New quantitative experimentsNew quantitative experiments (With Paul Umbanhowar)(With Paul Umbanhowar)