A swarm of small aerial drones performs a synchronised light display above a stadium
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Emergence and self-organisation: collective order from local rules
Problem
A swarm of small aerial drones performs a synchronised light display above a stadium. Overhead camera frames divide the sky into a grid of equal cells; each cell contains a drone with probability p=0.20, independently of every other cell, as in site percolation.
((a)) The mean occupancy of a block of n independent cells has standard error σ=p(1−p)/n. Find the least number of cells n that must be averaged together to measure the mean occupancy to a precision of ±0.020.
((b)) As a reference, suppose the drones' headings were instead independent random directions (a fully disordered state). For N such random unit vectors in the plane, the alignment order parameter A=∣∑ie^i∣/N has root-mean-square value ⟨A2⟩=N−1/2. A block of the size found in part (a) holds on average pn drones. Taking N=pn, compute this random baseline value of ⟨A2⟩.
((c)) Five independent runs of the actual display, with N drones each (different random seeds, each run allowed to reach steady state with the first half of its steps discarded), give alignment values A=0.62,0.58,0.65,0.60,0.59. Compute the mean of these five values and its standard error (the sample standard deviation divided by 5), and state whether the result is evidence of genuine collective alignment or is consistent with the random baseline of part (b).