Minimal stochastic field equations for one-dimensional flocking

TL;DR

This paper derives minimal stochastic field equations with local interactions that accurately reproduce the flocking behavior of active particles in one dimension, including stochastic direction changes, without long-range interactions.

Contribution

The authors develop a set of local-interaction stochastic field equations that replicate one-dimensional flocking behavior observed in agent-based models, highlighting the role of multiplicative noise.

Findings

Stochastic field equations reproduce flocking and alternating phases.

A multiplicative voter-model type noise is essential.

Nonlinear cubic alignment interaction is necessary for flocking.

Abstract

We consider the collective behaviour of active particles that locally align with their neighbours. 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 behaviour 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 behaviour 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 characterise 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.