A swarm of small underwater robots is modelled with the Vicsek model: each robot…

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 swarm of

N=900

small underwater robots is modelled with the Vicsek model: each robot moves at fixed speed

v_0

and turns towards the mean heading of neighbours within radius

r_0

, with angular noise amplitude

\eta

. Its order parameter is

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

. The swarm is run at high noise,

\eta=5

(out of a maximum of

2\pi

), for

4000

time steps, with the first

2000

steps discarded as the transient. ((a)) The instantaneous value of

v_{\rm a}

recorded at each of the remaining

2000

steps has a standard deviation of

0.018

, and successive values stay correlated for about

40

steps, so the sampled window contains only about

2000/40

effectively independent values. Estimate the standard error on the time-averaged

v_{\rm a}

. ((b)) Compute the random baseline value of

v_{\rm a}

expected for

N=900

headings pointing in independent random directions (the fully disordered limit), using

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

. ((c)) The time-averaged order parameter measured for this run is

v_{\rm a} = 0.075

. State, with the number of standard errors involved, whether independent random headings can explain this value. Then say whether this one run shows that the swarm is ordered, and what further runs would decide it.

Browse the problem library