A continuous-time persistent random walk model for flocking

TL;DR

This paper introduces a continuous-time persistent random walk model for active particles, capturing flocking behavior, clustering, and complex spatial states through bifurcation analysis and numerical simulations.

Contribution

It presents a novel model that incorporates interactions among persistent random walkers, leading to new insights into flocking and clustering phenomena.

Findings

Model exhibits transitions to flocking, clustering, and complex states.

Bifurcation analysis explains the emergence of collective behaviors.

Numerical simulations validate theoretical predictions.

Abstract

Random walkers characterized by random positions and random velocities lead to normal diffusion. A random walk was originally proposed by Einstein to model Brownian motion and to demonstrate the existence of atoms and molecules. Such a walker represents an inanimate particle driven by environmental fluctuations. On the other hand, there are many examples of so-called "persistent random walkers", including self-propelled particles that are able to move with almost constant speed while randomly changing their direction of motion. Examples include living entities (ranging from flagellated unicellular organisms to complex animals such as birds and fish), as well as synthetic materials. Here we discuss such persistent non-interacting random walkers as a model for active particles. We also present a model that includes interactions among particles, leading to a transition to flocking, that is, to a net flux where the majority of the particles move in the same direction. Moreover, the model exhibits secondary transitions that lead to clustering and more complex spatially structured states of flocking. We analyze all these transitions in terms of bifurcations using a number of mean field strategies (all to all interaction and advection-reaction equations for the spatially structured states), and compare these results with direct numerical simulations of ensembles of these interacting active particles.