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

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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…


Intermittency and clustering in a system of self driven particles

…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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