Mean-field theory of collective motion due to velocity alignment

TL;DR

This paper develops a mean-field theoretical framework for collective motion in self-propelled agents with velocity alignment, analyzing how individual dynamics influence the onset and nature of collective behavior.

Contribution

It introduces a mean-field model derived from microscopic dynamics, highlighting the impact of propulsion functions on the transition to collective motion and analyzing the effective temperature dynamics.

Findings

The type of propulsion function affects whether the transition to collective motion is continuous or discontinuous.

The effective temperature decreases significantly with increasing collective motion, despite constant individual fluctuations.

Mean-field measurements may misrepresent individual fluctuation behavior, cautioning their interpretation.

Abstract

We introduce a system of self-propelled agents (active Brownian particles) with velocity alignment in two spatial dimensions and derive a mean-field theory from the microscopic dynamics via a nonlinear Fokker-Planck equation and a moment expansion of the probability distribution function. We analyze the stationary solutions corresponding to macroscopic collective motion with finite center of mass velocity (ordered state) and the disordered solution with no collective motion in the spatially homogeneous system. In particular, we discuss the impact of two different propulsion functions governing the individual dynamics. Our results predict a strong impact of the individual dynamics on the mean field onset of collective motion (continuous vs discontinuous). In addition to the macroscopic density and velocity field we consider explicitly the dynamics of an effective temperature of the agent system, representing a measure of 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 {\em et al.}, PNAS, 106 (14), 2009].