D62 Analysis of emerging patterns

Topic

This analysis of emerging patterns derives the swarm's ultimate collective property from the individual trajectories of its members—thereby establishing another model for the system—in search of the signature of self-organization, criticality, or a specific emergent phenomenon within the swarm.

The swarm's final polar order parameter distinguishes between an ordered swarm—where individuals fly in a shared direction—and a disordered swarm, characterized by essentially random movement among its members. As the swarm aligns more closely with this common flight direction, its polar order parameter approaches its maximum value.

The swarm's final rotation parameter detects vortex or toroidal patterns within the group, while the radial distribution function measures the probability of finding a neighbor at a specific distance—relative to the probability expected from a completely random distribution of individuals. A distinct peak in this radial distribution function reveals a preferred segregation distance between two individuals—a specific personal space—corresponding to the concept of an interaction radius, which constitutes yet another model for the system. The swarm's correlation length constitutes perhaps the most informative observable of the entire analysis: if this length is significantly greater than the average spacing between individuals, the swarm exhibits long-range correlation among its members and responds collectively to any specific disturbance—far beyond the local interaction radius established by other models. The fact that the correlation length greatly exceeds the average spacing between individuals results in an even more coordinated collective response to disturbances.

Near a critical point, the swarm exhibits a correlation length that grows without bound, accompanied by massive fluctuations in the number of individuals—fluctuations far larger than those found in a completely random distribution of the same individuals. As the swarm approaches this critical point, these fluctuations in the number of individuals become even more massive.

Consequently, this model—incorporating the swarm's collective property derived from the individual trajectory mentioned at the start—demonstrates that the speed of information propagation within the swarm can far exceed the individual flight speed of any single member. This explains how the swarm responds to a predator much faster than an individual under attack could, on its own, alert its immediate neighbors. The fact that this final speed of information propagation within the swarm turns out to be even greater than the individual flight speed itself makes the swarm's final collective property of self-organization—with which this model began—even more evident.

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