Title:

Nonequilibrium emergent interactions between runandtumble random walkers

Nonequilibrium statistical physics involves the study of manyparticle systems that break time reversibilityalso known as detailed balanceat some scale. For states in thermal equilibrium, which must respect detailed balance, the comprehensive theory of statistical mechanics was developed to explain how their macroscopic properties arise from interactions between their microscopic constituent particles; for nonequilibrium states no such theory exists. The study of active matter, made up of particles that individually transduce free energy to produce systematic movement, provides a paradigm in which to develop an understanding of nonequilibrium behaviours. In this thesis, we are interested in particular in the microscopic interactions that generate the clustering of active particles that has been widely observed in simulations, and may have biological relevance to the formation of bacterial assemblages known as biofilms, which are an important source of human infection. The focus of this thesis is a microscopic latticebased model of two random walkers interacting under mutual exclusion and undergoing the runandtumble dynamics that characterise the motion of certain species of bacteria, notably Escherichia coli. I apply perturbative and exact analytic approaches from statistical physics to three variants of the model in order to find the probability distributions of their nonequilibrium steady states and elucidate the emergent interactions that manifest. I first apply a generating function approach to the model on a onedimensional periodic lattice where the particles perform straight line runs randomly interspersed by instantaneous velocity reversals or tumbles, and find an exact solution to the stationary probability distribution. The distribution can be interpreted as an effective nonequilibrium pair potential that leads to a finiterange attraction in addition to jamming between the random walkers. The finiterange attraction collapses to a delta function in the limit of continuous space and time, but the combination of this jamming and attraction is suffciently strong that even in this continuum limit the particles spend a finite fraction of time next to each other. Thus, although the particles only interact directly through repulsive hardcore exclusion, the activity of the particles causes the emergence of attractive interactions, which do not arise between passive particles with repulsive interactions and dynamics respecting detailed balance. I then relax the unphysical assumption of instantaneous tumbling and extend the interacting runandtumble model to incorporate a finite tumbling duration, where a tumbling particle remains stationary on its site. Here the exact solution for the nonequilibrium stationary state is derived using a generalisation of the previous generating function approach. This steady state is characterised by two lengthscales, one arising from the jamming of approaching particles, familiar from the instant tumbling model, and the other from one particle moving when the other is tumbling. The first of these lengthscales vanishes in a scaling limit where continuum dynamics is recovered. However, the second, entirely new, lengthscale remains finite. These results show that the feature of a finite tumbling duration is relevant to the physics of runandtumble interactions. Finally, I explore the effect of walls on the interacting runandtumble model by applying a perturbative graphtheoretic approach to the model with reflecting boundaries. Confining the particles in this way leads to a probability distribution in the low tumble limit with a much richer structure than the corresponding limit for the model on a periodic lattice. This limiting probability distribution indicates that an interaction over a finite distance emerges not just between the particles, but also between the particles and the reflecting boundaries. Together, these works provide a potential pathway towards understanding the clustering of selfpropelled particles widely observed in active matter from a microscopic perspective.
