A continuous-time persistent random walk model for flocking

A classical random walker is characterized by a random position and velocity. This sort of 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. Interacting self-propelled particles have the potential to exhibit a number of self-coordinated motions. Nature offers many examples surprising for their beauty, such as flocking birds or swarming fish. The keys to understanding the emergence of such collective behaviors are two: the motion of the self-propelled entities themselves and the interaction that leads to the coordination. In this work, we present a mathematical model for the sort of self-propelled particles that under appropriate conditions are capable of collective motions. This model deepens our understanding of the emergence of collective motion in terms of the theoretical framework provided by nonequilibrium statistical mechanics and nonlinear physics.D.E. thanks FONDECYT project No. 1170669 for financial support. R.T. acknowledges financial support from Ministerio de Economía y Competitividad (MINECO) and Fondo Europeo de Desarrollo Regional (FEDER) under project ESOTECOS FIS2015-63628-C2-1-R. D.E. and R.T. acknowledge the warm hospitality at UCSD, where most of this work was carried out.Peer reviewe