Nonequilibrium emergent interactions between run-and-tumble random walkers

Nonequilibrium statistical physics involves the study of many-particle systems that break time reversibility|also known as detailed balance|at 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 lattice-based model of two random walkers interacting under mutual exclusion and undergoing the run-and-tumble 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 one-dimensional 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 non-equilibrium pair potential that leads to a finite-range attraction in addition to jamming between the random walkers. The finite-range 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 hard-core 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 run-and-tumble 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 run-and-tumble interactions. Finally, I explore the effect of walls on the interacting run-and-tumble model by applying a perturbative graph-theoretic 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 self-propelled particles widely observed in active matter from a microscopic perspective

X-ray and Laser Investigation of High Pressure Sprays

The diagnostics of multiphase flows is important to understand the correlation of parameters such as nozzle geometry, flow velocity, liquid breakup characteristics, atomization, and mass distribution. Investigation of each of these parameters can lead to various improvements on design and optimization, for example, of combustion in gas turbines and rocket engines. For this project, the X-ray and optical diagnostics of multiphase flows in propulsion is investigated for high pressures. Two x-ray tube sources are used to illuminate the spray, and the images are captured on phosphor plates coupled to high-speed cameras. Reconstruction of the mass distribution is accomplished using computer analysis by applying Beer’s-Lambert law. High temporal resolution and spatial resolution images were collected of various sprays composed of water and KI solutions for development and characterization of the technique with different window materials typically used in high-pressure vessels. Laser-based fluorescence was tested in vaporized jet-A fuel in a high-pressure vessels. The implications for X-ray and laser analysis of multiphase flows in sprays with high spatial and temporal resolution and high signal to noise ratio are investigated

Optimization of a High-Speed X-Ray Imaging System for Studying Sprays

Spray-based liquid atomization and liquid mixing is critical for development of efficient combustors and drug delivery systems as well as multiple coating-related applications. While optical methods allow characterization of low density regions of sprays, the scattering of optical photons hinders the characterization of a dense core. Unlike optical photons, higher-energy X-ray photons have the capability to penetrate and image the core structure of sprays. Here we characterized temporal and spatial resolution of an X-Ray imaging system based on a commercially available tube source with an anode size of 0.6 mm. For high-speed imaging, a phosphor screen in combination with a high-speed CMOS camera equipped with a two-stage intensifier was used. Water was used as a model liquid with the addition of potassium iodide to increase the X-Ray absorption coefficient. Two-dimensional images of 0.5mm and 2 mm impinging jet sprays were taken with differing spatial resolutions and potassium iodide mass concentrations. Depending on the spray conditions, optimal imaging settings were found. The technique can be extended to three-dimensional analysis of sprays with multiple viewing angles from two or more X-ray sources along with tomographic reconstruction

Jamming and Attraction of Interacting Run-and-Tumble Random Walkers

We study a model of bacterial dynamics where two interacting random walkers perform run-and-tumble motion on a one-dimensional lattice under mutual exclusion and find an exact expression for the probability distribution in the steady state. This stationary distribution has a rich structure comprising three components: a jammed component, where the particles are adjacent and block each other; an attractive component, where the probability distribution for the distance between particles decays exponentially; and an extended component in which the distance between particles is uniformly distributed. The attraction between the particles is sufficiently strong that even in the limit where continuous space is recovered for a finite system, the two walkers spend a finite fraction of time in a jammed configuration. Our results potentially provide a route to understanding the motility-induced phase separation characteristic of active matter from a microscopic perspective.Comment: 8 pages, 3 figure

Exact Solution of Two Interacting Run-and-Tumble Random Walkers with Finite Tumble Duration

We study a model of interacting run-and-tumble random walkers operating under mutual hardcore exclusion on a one-dimensional lattice with periodic boundary conditions. We incorporate a finite, Poisson-distributed, tumble duration so that a particle remains stationary whilst tumbling, thus generalising the persistent random walker model. We present the exact solution for the nonequilibrium stationary state of this system in the case of two random walkers. We find this to be characterised by two lengthscales, one arising from the jamming of approaching particles, and the other from one particle moving when the other is tumbling. The first of these lengthscales vanishes in a scaling limit where the continuous-space dynamics is recovered whilst the second remains finite. Thus the nonequilibrium stationary state reveals a rich structure of attractive, jammed and extended pieces.Comment: 27 pages, 4 figures, Mathematica notebook as supplementary materia

Run and tumble particle under resetting:a renewal approach

We consider a particle undergoing run and tumble dynamics, in which its velocity stochastically reverses, in one dimension. We study the addition of a Poissonian resetting process occurring with rate $r$. At a reset event the particle's position is returned to the resetting site $X_r$ and the particle's velocity is reversed with probability $\eta$. The case $\eta = 1/2$ corresponds to position resetting and velocity randomization whereas $\eta =0$ corresponds to position-only resetting. We show that, beginning from symmetric initial conditions, the stationary state does not depend on $\eta$ i.e. it is independent of the velocity resetting protocol. However, in the presence of an absorbing boundary at the origin, the survival probability and mean time to absorption do depend on the velocity resetting protocol. Using a renewal equation approach, we show that the the mean time to absorption is always less for velocity randomization than for position-only resetting.Comment: 16 pages, 1 figure, version accepted in Journal of Physics

An experimental and modeled comparison of diffraction in imaging systems

The resolution limit of imaging systems is ultimately limited by diffraction. However, diffraction is often neglected in the analysis and design of both front and back illumination imaging systems in favor of the simpler ray tracing model. In many systems, paraxial optics provides a reasonable model for the design of systems with high resolution. This is certainly true for the majority of front-illuminated imaging systems; however, in back illuminated (shadowgraphic) imaging systems resolution is very strongly affected by diffraction. We present a detailed experimental comparison of imaging resolution differences between front and back illuminated imaging systems for non-scattering and scattering environments. Additionally, modeling results of both systems are compared with the experimental results and classical optical theory. Preliminary results and calculations show that physical optics creates a stronger effect on resolution in front illuminated systems in either scattering or non-scattering environments despite original predictions