D63 Measurement of spatial correlations

Topic

The measurement of spatial correlations within the swarm quantifies the extent to which an individual's behavior predicts that of its neighbors, based on both the distance separating them and the time elapsed between them.

The velocity correlation function serves as the primary observable in this analysis: if the function approaches its maximum value at distances significantly greater than the average spacing between individuals, the swarm exhibits the long-range correlation characteristic of this class of models.

In a critical swarm near the Vicsek phase transition, the velocity correlation function decays according to a power law rather than exponentially, with the correlation length growing without a defined limit. As the swarm approaches the Vicsek phase transition more closely, this correlation length increases further toward that unbounded growth.

The structure function complements the velocity correlation function introduced earlier; it describes how the velocity difference between a pair of individuals varies with the distance between them, and its scaling behavior with distance reveals the turbulence or correlation regime of the swarm. The energy spectrum of this swarm—within its specific frequency domain—constitutes an analogue to the Kolmogorov turbulence spectrum; consequently, this active, non-equilibrium swarm may exhibit a characteristic exponent that differs significantly from that of classical turbulence.

The swarm's susceptibility serves as an analogue to specific heat in statistical mechanics; the divergence of this susceptibility signals proximity to the critical point associated with giant fluctuations in the number of individuals. As the swarm approaches this critical point more closely, the susceptibility diverges even more intensely.

Thus, the model—which began by predicting neighbor behavior based on distance and time—distinguishes between a random walk (characterized by a short directional persistence time, derived from temporal autocorrelation) and directed motion (characterized by a significantly longer persistence time). A longer directional persistence time for an individual within the swarm results in a more accurate prediction of that individual's behavior over time.

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