Minimal stochastic field equations for one-dimensional flocking

Authored by E O Laighleis, M R Evans, R A Blythe

Date Published: 2018

DOI: 10.1103/physreve.98.062127

Sponsors: United Kingdom Engineering and Physical Sciences Research Council (EPSRC)

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

Abstract

We consider the collective behavior of active particles that locally align with their neighbors. Agent-based simulation models have previously shown that in one dimension, these particles can form into a flock that maintains its stability by stochastically alternating its direction. Until now, this behavior has been seen in models based on continuum field equations only by appealing to long-range interactions that are not present in the simulation model. Here, we derive a set of stochastic field equations with local interactions that reproduces both qualitatively and quantitatively the behavior of the agent-based model, including the alternating flock phase. A crucial component is a multiplicative noise term of the voter model type in the dynamics of the local polarization whose magnitude is inversely proportional to the local density. We show that there is an important subtlety in determining the physically appropriate noise, in that it depends on a careful choice of the field variables used to characterize the system. We further use the resulting equations to show that a nonlinear alignment interaction of at least cubic order is needed for flocking to arise.

Tags

Density System Phase-transition Motion