Interacting active particles and cellular automata: microscopic models of stochastic nonequilibrium systems

Abstract

The broad focus of this thesis is the statistical physics of nonequilibrium systems -- systems which violate the condition of detailed balance. Detailed balance is the property that the probability fluxes between all pairs of states of a system cancel exactly, and is the defining feature of the equilibrium stationary state. Nontrivial flux cancellations -- which arise as a result of probability currents in a system's state space -- are also possible: when this happens, a system is said to be in a equilibrium stationary state. As a consequence of this increased complexity, establishing a general theory of nonequilibrium systems remains one of physics' great outstanding challenges.

One route to understanding nonequilibrium systems is to consider simplified microscopic models. The benefits to this are twofold: firstly, microscopic models allow us to probe nonequilibrium processes at the most fundamental level; and secondly, by pooling together the results of various such models, we can begin to establish connections between them with a view to extracting general principles.

In this thesis, we will consider two kinds of nonequilibrium system when developing microscopic models. The first kind is active systems. Active systems -- those which comprise particles that can execute motion using energy they extract from their environment -- are inherently out of equilibrium. In the presence of interactions, active particles often exhibit intriguing emergent behaviours -- such as collective motion -- which are absent in systems of passive particles. Not only, therefore, do they hold strong scientific interest, but their ubiquity in nature provides a pool of inspiration for physicists from which models may be developed and studied.

The dearth in the literature of N-body microscopic models of active systems, where N ≥ 2, is a major problem if we wish to fundamentally understand how their macroscopic complexity emerges from interactions at the level of individual particles. In this thesis, we shall devote much time to the analysis of microscopic active systems in an attempt to address this. To this end, two models of interacting persistent random walkers (PRWs) will be discussed. The PRW is an active particle which moves by executing a series of linear `runs' at speed γ interspersed by random reorientation events, known as `tumbles', which occur at rate ω. Interacting PRW systems are of interest in the active-matter community as they are often inspired by and have applications to bacterial motion.

The first model consists of a dilute gas of excluding PRWs in an arbitrary number of dimensions. The jamming probability -- the probability of finding a pair of adjacent PRWs mutually blocking each other -- is derived via a first-passage approach in order to investigate the nature of multiparticle clustering. This yields the surprising result of a vanishing critical density for clustering in the model's thermodynamic limit.

The second PRW model generalises the exclusion interaction to an active \emph{recoil} interaction, whereby PRWs are instantaneously displaced upon contact according to an arbitrary distribution. The nonequilibrium stationary states of a one-dimensional two-PRW system featuring recoils are derived exactly using a generating-function approach, revealing a variety of rich behaviours. These include reentrant states, for which an effective attraction exists between the walkers within a finite range of tumble rates but is otherwise repulsive, as well as highly nontrivial boundary behaviours, for which a novel technique was developed in order for an exact characterisation to be established.

The second kind of nonequilibrium system we shall consider is the cellular automaton (CA). A CA consists of a lattice whose sites are associated with a discrete variable. For all sites, this variable is updated in discrete time steps according to a prescribed function whose input is the site's local environment. Thus, given some initial configuration, the CA evolves through a deterministic trajectory of configurations. Engineering the update function to enforce broken detailed balance allows for the possibility of complex emergent phenomena. The CA, therefore, constitutes an alternative approach for the nonequilibrium physicist regarding the examination of emergent complexity from microscopic rules for which detailed balance does not hold.

In the final chapter of this thesis, a one-dimensional CA of binary variables is modified such that two of its eight possible update rules are made stochastic. We then assess the changes this effects on the state space via simulations, revealing a pattern in the configurational structure of each sector (subspace). From this we establish a lower bound on the number of sectors of the model; we find that this lower bound is related to the Fibonacci numbers and asymptotically grows as (⁴√ϕ)ᴸ, where ϕ is the golden ratio and L is system size. This growth rate, it turns out, is substantially slower than the stochastic automaton's deterministic predecessor.

Taking together the above-described models and the results we derive from them, it is hoped that this thesis constitutes a small step forward in our understanding of nonequilibrium systems.