Mean-field theory of collective motion due to velocity alignment

Establishing a direct link between individual based models and the corresponding population description is a common challenge in theoretical ecology. Swarming is a prominent example, where collective effects arising from interactions of individuals are essential for the understanding of large-scale spatial population dynamics, and where both levels of modeling have been often employed without establishing this connection.Here, we consider a system of self-propelled agents with velocity alignment in 2D and derive a mean-field theory from the microscopic dynamics via a nonlinear Fokker–Planck equation and a moment expansion of the probability density. We analyze the stationary solutions corresponding to macroscopic collective motion (ordered state) and the disordered solution with no collective motion in the spatially homogeneous system. In particular, we discuss the impact of different propulsion functions governing individual dynamics. Our results predict a strong impact of individual dynamics on the mean field onset of collective motion (continuous vs discontinuous). In addition to the macroscopic density and velocity fields, we consider the effective “temperature” field, measuring velocity fluctuations around the mean velocity. We show that the temperature decreases strongly with increasing level of collective motion despite constant fluctuations on individual level, which suggests that extreme caution should be taken in deducing individual behavior, such as, state-dependent individual fluctuations from mean-field measurements (Yates et al., 2009).

► Systematic derivation of coarse-grained equations from individual dynamics.
► Analysis of the onset of collective motion in systems of active Brownian agents.
► Uncovering the impact of individual velocity dynamics on the mean-field solutions.