Intermittency and clustering in a system of self-driven particles

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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

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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

- Model background
- Self-driven particle model (SDPM)
- Dynamical phase transition

- Intermittency
- Numerical evidence
- Two-body problem solution

- Clustering
- Cluster dynamics
- Cluster statistics

- Conclusion

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 by Vicsek et al.
At every t we update using

- Order parameter

The ordered phase

- For , the particles align.
- Simulation parameters:
- =1
- =1000
- =0.1
- = 0.8
- = 0.4

Ordered phase appears because of long-range interactions over time

- 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.

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

Signature of

intermittency

PDF of

Histogram of laminar intervals

Intermittent signal in time

- 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.

- 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.

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

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

- 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