Stochastic effects in systems of aligning self-propelled particles

Abstract

Systems of self-propelled particles are often capable of exhibiting complex behaviours on a macroscopic scale with only simple interactions between the active microscopic agents. In systems where the particles interact by attempting to align their directions of motion, the ordered steady state tends towards a dense coherent grouping of particles (a "flock") travelling in the same direction. Continuum theories where the particles are treated as an active fluid allow for a greater understanding of the macroscopic behaviour of these flocks, although these theories have typically focussed on understanding the behaviour of the flock in a steady state. In this thesis, we are interested in deriving continuum theories of aligning self-propelled particle systems and in understanding the role that stochasticity has in ensuring the dynamical behaviour of the underlying agentbased model is maintained.

The focus of this thesis is a family of models of aligning self-propelled particles on a lattice that interact by aligning with a subset of nearby particles. We use the Kramers-Moyal approximation to derive stochastic Langevin equations directly from the microscopic interactions. Our goal is to obtain equations for the evolution of the system's density and polarisation such that their trajectories match the dynamic behaviour of the underlying agent-based models and, in doing so, to demonstrate that the form of the stochastic prefactor in the polarisation equation can greatly affect the macroscopic behaviour of the system.

In Chapter 2, we study the ordered state of a system of aligning self-propelled particles on a one-dimensional lattice. The aligning interaction between particles allows for the formation of a flock capable of alternating the direction it travels through the lattice. We derive a set of stochastic differential equations for the density and polarisation of the system and introduce a numerical integration scheme to demonstrate that the order parameter of each of the agent-based and continuum systems scales identically with increasing noise strength. We then use the continuum equations to obtain a minimal set of interactions for a flock to exist in one dimension and demonstrate how alignment interactions with three particles are necessary for a flock to form on a one dimensional lattice.

This motivates the work in the remainder of this thesis, wherein we examine a family of two-dimensional models to explore whether we can derive stochastic differential equations whose trajectories demonstrate the same behaviour as in the agent-based models. We introduce a family of four lattice-based agent-based models in Chapter 3 and map out the behaviour of the ordered state in each of these models. These models consist of all combinations of two interaction types (exponential or linear in local polarisation) and two interaction neighbourhoods (fi xed or varying with local density). One of these models shows the "banding" present in traditional Vicsek models, while the other three show the occasional macroscopic change in direction observed in flocks of birds such as starlings.

In Chapter 4, we use the Kramers-Moyal approximation again to derive stochastic differential equations for the density and polarisation of the four models above. Using linear stability analysis, we explain why the ordered state in each model consists of a flock that will either be capable or incapable of turning. The linear stability analysis shows why the choice of interaction neighbourhood does not affect the ability of the flocks with a linear interaction to macroscopically alter direction and why that choice does affect the ability of a flock to turn for systems with an exponential interaction, although some calculation remains here to demonstrate linear stability exactly matching that of the agent-based systems. We also explore a numerical integration scheme for the two dimensional models, laying out a procedure that may result in integrated trajectories matching the behaviour of the agent-based model as in one dimension.

The work in this thesis explores the effects of different stochastic terms in continuum equations describing systems of aligning self-propelled particles and introduces a mechanism to derive these terms to ensure the behaviour matches that of the underlying agent-based models. We demonstrate the power of this mechanism by identifying a minimal model of flocking in one dimension and by exploring when flocks can turn in two dimensions. These examples provide a pathway for exploring the dynamic behaviour of other interacting particle models on a macroscopic scale.