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

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

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

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

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

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

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