Intermittency and clustering in a system of self-driven particles Cristian Huepe Northwestern University Maximino Aldana University of Chicago Featuring valuable discussions with Hermann Riecke Mary Silber Leo P. Kadanoff

Outline Model background Intermittency Clustering Conclusion Self-driven particle model (SDPM) Dynamical phase transition Intermittency Numerical evidence Two-body problem solution Clustering Cluster dynamics Cluster statistics Conclusion

Model background Model by Vicsek et al. Order parameter At every t we update using Order parameter Angle of the velocity of the ith particle Random var. with constant distribution: Sum over all particles within interaction range r Periodic LxL box All particles have:

Dynamical phase transition The ordered phase For , the particles align. Simulation parameters: =1 =1000 =0.1 = 0.8 = 0.4

2D phase transition in related models Simulation parameters: = 20000 = 10 = 0.01 = 15 Analogous transitions shown R-SDPM: Randomized Self-Driven Particle Model VNM: 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. Ordered phase appears because of long-range interactions over time

Intermittency The real self-driven system presents an intermittent behavior Simulation parameters = 1000 = 0.1 = 1 = 0.4

Numerical evidence Intermittent signal in time PDF of Signature of intermittency PDF of Intermittent signal in time Histogram of laminar intervals

Two-body problem solution Two 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…

…the analytic result is obtained after a Laplace transform: … Computing the inverse Laplace transform, we compare our analytic approximation with the numerical simulations.

Clustering 2-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

Size transition statistics Mainly looses/gains few particles Detailed balance! Same power-law behavior for all sizes

Conclusion Intermittency appears in the ordered phase of a system of self-driven particles The intermittent behavior for a reduced 2-particle system was understood analytically The many-particle intermittency problem is related to the dynamics of clusters, which have: Scale-free sizes and size-transition probabilities Size transitions obeying detailed balance ………FIN