title: Collective Dynamics of Large Scale Multiagent Systems with Nonlocal Interactions
Creators: name: Kruk, Nikita
Publisher: name: UNSPECIFIED
Year: 2020

description: The thesis investigates the emergence of collective behavior in large scale multiagent systems. Examples of such dynamics are ample. The most familiar of them are such macroscale phenomena as flocking of birds, schooling of fish, and swarming of insects. The other common macroscale manifestations of collective dynamics include herds of sheep, human crowds, and robotic swarms etc. On the microscale, we often find ordered motion in such living systems as bacterial suspensions and cell layers as well as in such nonliving systems as shaken granular rods and colloidal particles. One of the fascinating features of all the aforementioned systems is that they consist of constituents of completely different nature but still do produce qualitatively similar behavior as the whole. However, those constituents have one common characteristic, namely, they are able to propel themselves either by using an intrinsic source of energy or by consuming it from the surrounding environment. Such systems are generically referred to as active matter. Our approach to the study of active matter is primarily based on theories of nonlinear dynamical systems and statistical physics. In our research, we put particular emphasis on nonlocality of interactions. Namely, these are the interactions that are defined on intermediate ranges with respect to the system size, in contrast to closest-neighbor and all-to-all interactions. The first part of the thesis is concerned with the analysis of active matter systems in terms of finite size particle models. Each particle serves as an abstraction for an underlying constituent, e.g., one of those described above, and it can be imagined as a collection of variables that describe the relevant information about it, e.g., a position, velocity, orientation etc. Then, the motion of a particle is described with nonlinear differential equations. At this point, the distinction should be made. First, the resulting collective behavior may be a consequence of intrinsic deterministic nonlinear dynamics themselves. In this case, particle's motion is described with ordinary differential equations and we employ the tools of nonlinear dynamical systems to characterize the resulting motion. It involves the description of fixed points, periodic orbits, and attractors, each of which corresponds to a specific collective motion pattern, as well as bifurcations that provide us with their stability. Second, collective phenomena in natural environment are inevitably subject to various internal and external perturbations. To take that into account, equations of particle motion are generalized to include stochastic forces. This way, particles are modeled as stochastic processes and their temporal evolution is described with stochastic differential equations. The important question in the theory of active matter is the stability of collective motion against noise and phase transitions between various collective motion patterns. The second part of the thesis discusses the continuum description of large scale interacting multiparticle systems. Namely, when the number of particles becomes large, the description of their dynamics in terms of finite size particle models becomes inefficient. To find a more efficient system description in such cases, we observe that when the number of particles grows, they cover the domain of interest finer and under an appropriate scaling, they will cover all of it in the limit of infinite population size. We realize this by assuming a statistical approach according to which we determine the probability to find a particle at a particular point of the phase space at a particular time. In such a limit, the collective motion of interacting multiparticle systems is described with (probability) density functions or, more generally, with (probability) measures. Their temporal evolution is described with partial differential equations, which, in the context of interacting multiparticle systems, involve nonlinear integral terms. There are two types of continuum descriptions of active matter that we employ. The first one is the kinetic one. It is concerned with the description of a system in terms of a density function that depends on all phase space variables. The second type of continuum description is the hydrodynamic one. This theory describes an active matter system with a set of a small number of fields that depend on spatial information only, in contrast to the kinetic theory. For example, one often considers fields that describe spatial distribution of particles, momenta, polar, or nematic orders. The third part of the thesis is about the development of numerical schemes, dedicated to continuum limit equations of interacting multiparticle systems. These are nonlinear partial integro-differential equations that require dedicated analysis due to their complexity. The numerical schemes, we developed, consider such properties of the approximate solution as positivity preservation, physical conservations, and free energy dissipation. With such a numerical scheme at hand, we study true continuum limit behavior of original finite size particle models as well as related phase transitions without finite size effects. The fourth part of the thesis covers the experimental work on the swarming of \textit{B. subtilis} as a controlled system of \textit{in vitro} active matter. This part includes preparation and execution of experiments, image processing, multitarget tracking, and mathematical modeling of a kind described above. We establish protocols for the swarming of \textit{B. subtilis} in microfluidic polydimethylsiloxane channels as well as on agar plates, which are straightforward to reproduce. This results in a sequence of images per experiment, which we afterwards expose to multitarget tracking algorithms to retrieve each bacterium's trajectory. This knowledge allows us to construct models that provide collective behavior closer to the natural setup. Unlike the standard active matter models, the description of bacterial swarming in a confined environment requires the consideration of bacterium's physical shape. This is often achieved by modeling a bacterium as a self-propelled rod.