intermittency and clustering in a system of self driven particles
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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

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

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
dynamical phase transition
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
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.
intermittency
Intermittency
  • The real self-driven system presents an intermittent behavior
  • Simulation parameters
    • = 1000
    • = 0.1
    • = 1
    • = 0.4
numerical evidence

Signature of

intermittency

PDF of

Histogram of laminar intervals

Numerical evidence

Intermittent signal in time

two body problem solution
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…
slide9

…the analytic result is obtained after a Laplace transform:

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

clustering
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
Cluster size statistics (particle number)
  • Power-law cluster size distribution (scale-free)
  • Exponent depends on noise and density
size transition statistics
Size transition statistics
  • Mainly looses/gains few particles
  • Detailed balance!
  • Same power-law behavior for all sizes
conclusion
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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