A model fish school of fish is simulated with the Vicsek model at noise amplitude (a…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Complex and collective systems

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

Problem

A model fish school of

N=1600

fish is simulated with the Vicsek model at noise amplitude

\eta=5

(a strongly disordered regime), with

L=20

r_0=1

v_0=0.03

, run for

4000

steps with the first

2000

discarded as the transient. Five independent seeds give the following time-averaged order parameters

v_{\rm a} = |\sum_i\mathbf v_i|/(Nv_0)

0.0316

0.0305

0.0300

0.0315

0.0304

. Runs at the same density and noise with

N=100

and

N=400

give

v_{\rm a} \approx 0.124

and

0.062

. ((a)) Treating the five seeds as independent blocks, compute the mean order parameter and its standard error. ((b)) Compute the random baseline value of

v_{\rm a}

expected for

N=1600

headings pointing in independent random directions, using

\langle A\rangle \approx \sqrt{\pi}/(2\sqrt{N})

. ((c)) How many independent seeds, in total, would be needed to bring the standard error on the mean below

0.0001

, assuming each seed's order parameter has the same spread as measured here? ((d)) Give the number of standard errors by which the five-seed mean exceeds the random baseline. Using the three system sizes, decide whether this excess shows an ordered school or short-range alignment among neighbours in a disordered one.

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