Write a simulation of the Vicsek model as defined in the text: particles at fixed…

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Thermal, Statistical & Complex Systems · Monte Carlo and lattice simulation of physical systems

Many Parts, One Behaviour, Ch Emergence and self-organisation: collective order from local rules

Problem

Write a simulation of the Vicsek model as defined in the text:

N

particles at fixed speed

v_0 = 0.03

in a periodic square box of side

L

; at each step every particle takes the direction of the summed unit headings of all particles within

r_0 = 1

of it, itself included, adds a noise

\eta\xi

with

\xi

uniform on

[-\tfrac12,\tfrac12]

, and moves

v_0

along the new heading. Use density

N/L^2 = 4

, with

N = 100

(

L = 5

) and

N = 400

(

L = 10

), random initial positions and headings,

4000

steps with the first

2000

discarded as transient, and seeds

1

5

. Your digits will differ from any other program's, because random-number generators and draw orders differ; what is assessed is the pattern with

N

and

\eta

and whether your error bars support it. ((a)) Compute the random baseline

\sqrt{\pi}/(2\sqrt N)

for each size. ((b)) For each size, report the time-averaged

v_{\rm a}

\eta = 1

3

and

5

as the mean over the five seeds with its standard error. ((c)) Say how the results change with

N

on the ordered and the disordered side, and whether the values at

\eta = 5

show order.

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