As a control run before switching on the alignment rule

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

As a control run before switching on the alignment rule, a researcher initialises N=64N = 64 particle headings uniformly at random in a Vicsek-style simulation and measures the alignment order parameter va=∣∑ie^i∣N=0.13.v_{\rm a} = \frac{\left|\sum_i \hat{\mathbf e}_i\right|}{N} = 0.13. They conclude: "Even with the alignment rule switched off, the flock already shows a noticeable 13%13\% order, so collective ordering must be emerging spontaneously from the boundary conditions alone." ((a)) Find the error in this reasoning. ((b)) Give the value that ⟨va2⟩\sqrt{\langle v_{\rm a}^2\rangle} takes for NN headings pointing in independent random directions, and use it to judge whether 0.130.13 is evidence of order.

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