# Intermittency and clustering in a system of self-driven particles - PowerPoint PPT Presentation

1 / 14

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)

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Intermittency and clustering in a system of self-driven particles

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## 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

### Outline

• 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 background

• Model by Vicsek et al.

At every t we update using

• Order parameter

### Dynamical phase transition

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

### 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

• 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

### Numerical evidence

Intermittent signal in time

### 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…

…the analytic result is obtained after a Laplace transform:

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

### 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)

• Power-law cluster size distribution (scale-free)

• Exponent depends on noise and density

### Size transition statistics

• Mainly looses/gains few particles

• Detailed balance!

• Same power-law behavior for all sizes

### 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