Intermittency and clustering in a system of self-driven

particles

Cristian HuepeNorthwestern University

Maximino AldanaUniversity of Chicago

Featuring valuable discussions with

•Hermann Riecke•Mary Silber•Leo P. Kadanoff

Outline

Model backgroundSelf-driven particle model (SDPM)Dynamical phase transition

IntermittencyNumerical evidenceTwo-body problem solution

ClusteringCluster dynamicsCluster statistics

Conclusion

Model by Vicsek et al. At every t we update

using

Order parameter

Model background

ttvtt irxxji

ij

||

Angle)(

tttvtxttx iii

dttvvNT

N

ii

T

NT

10 0

11lim

),( ii vx

Angle of the velocity of the ith particle

Sum over all particles within interaction range r

•Periodic LxL box•All particles have: 0vvi

2,2

i

Random var. with constantdistribution:

For , the particles align.Simulation parameters:

=1 =1000 =0.1 = 0.8 = 0.4

Dynamical phase transition

2LN

c

rN0v

The ordered phase

2D phase transition in related models

Ordered phase appears because of long-range interactions over time

Simulation parameters: = 20000 = 10 = 0.01 = 15

Analogous transitions shown

R-SDPM: Randomized Self-Driven Particle ModelVNM: Vectorial Network Model Link pbb to random element: 1-p Link pbb to a K nearest neighbor: p

Analytic solution found for VNM with p=1.

0v

N

tK i

Intermittency

The real self-driven system presents an intermittent behaviorSimulation parameters

= 1000 = 0.1 = 1 = 0.4

N0v

Numerical evidence

Intermittent signal in time

PDF of t

Histogram of laminar intervals

Signature ofintermittency

2

2

21

12arccos4

P

Two-body problem solutionTwo states: Bound (laminar) & unbound (turbulent).Intermittent burst = first passage in (1D) random walk

Average random walk step size = Continuous approximation: Diffusion equation with

Solving simple 1D problem for the Flux at x=r with one absorbing and one reflecting boundary condition…

2

2 ,,

x

txcD

t

txc

tD 2/2

rx

xx

… Computing the inverse Laplace transform, we compare our analytic approximation with the numerical simulations.

r

rx

x

xsj

sxcDxsj

D

scosh

D

scosh

0

00

0

;

),(;

…the analytic result is obtained after a Laplace transform:

Clustering2-particle analysis to N-particles by defining clusters.Cluster = all particles connected via bound states.Clusters present high internal order.Bind/unbind transitions = cluster size changes.

Cluster size statistics (particle number)

Power-law cluster size distribution (scale-free)Exponent depends on noise and density

Mainly looses/gains few particlesDetailed balance!Same power-law behavior for all sizes

Size transition statistics

Conclusion

………FIN

Intermittency appears in the ordered phase of a system of self-driven particlesThe intermittent behavior for a reduced 2-particle system was understood analyticallyThe many-particle intermittency problem is related to the dynamics of clusters, which have:

Scale-free sizes and size-transition probabilitiesSize transitions obeying detailed balance