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Intermittency and clustering in a system of self-driven particles

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 Self-driven particle model (SDPM)

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Intermittency and clustering in a system of self-driven particles

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  1. 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

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

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

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

  5. Ordered phase appears because of long-range interactions over time 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.

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

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

  8. 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…

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

  10. 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.

  11. Cluster size statistics (particle number) • Power-law cluster size distribution (scale-free) • Exponent depends on noise and density

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

  13. 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

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