Transcript bas2003 3868

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)
Dynamical phase transition
Intermittency
Numerical evidence
Two-body problem solution
Clustering
Cluster dynamics
Cluster statistics
Conclusion
Model background
Model by Vicsek et al.
 
(
x
At every t we update i , vi ) using



xi t  t   xi t   vi t  t t

 
 i (t  t )  Angle   v j t   i t 
| x j  xi | r 
Angle of the
velocity of the
ith particle
Sum over all
particles within
interaction range r
Order parameter
  lim
T 
N 
1
NT
T
1
0 v0
Random var.
with constant
distribution:
 i    2 , 2 



 vi t  dt
N
i 1
•Periodic LxL box 
•All particles have: vi  v0
Dynamical phase transition
The ordered phase
For   c , the
particles align.
Simulation
parameters:
r =1
N =1000
v0 =0.1

= 0.8
N
L2
= 0.4

2D phase transition in related models
Ordered phase appears
because of long-range
interactions over time
Simulation parameters:
= 20000
 = 10
v0 = 0.01
Ki t  = 15
N
Analogous transitions shown
R-SDPM: Randomized SelfDriven 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 selfdriven system
presents an
intermittent
behavior
Simulation
parameters
N = 1000
v0 = 0.1
 =1
 = 0.4

Numerical evidence

Signature of
intermittency

4   arccos 2  2  1 
P   2 

2
 
1 

PDF of t 
Intermittent signal in time
Histogram of laminar intervals
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 D   2 / 2t
cx, t 
 2 c  x, t 
D
t
x 2
Solving simple 1D problem for the
Flux at x=r with one absorbing and
one reflecting boundary condition…
x  x
r
x
…the analytic result is obtained after a Laplace transform:
j s; x0   Dc( x, s ) x 0
j s; x0  

cosh 


s
 x0  r  
D


cosh 

s 
r
D 
… 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