Intermittency and clustering in a system of selfdriven 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

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Outline

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

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Model background

• Model by Vicsek et al.
• At every t we update using
• Order parameter

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Dynamical phase transition
The ordered phase

• For , the particles align.
• Simulation parameters
• 1
• 1000
• 0.1
• 0.8
• 0.4

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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 p1.

• Ordered phase appears because of long-range
  interactions over time

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Intermittency

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

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Numerical evidence
Intermittent signal in time
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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 xr
  with one absorbing and one reflecting boundary
  condition

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

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Cluster size statistics (particle number)

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

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Size transition statistics

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

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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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