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The statistical mechanics of a one-dimensional Ising model in thermal equilibrium is well-established, textbook material. Yet, when driven far from equilibrium by coupling two sectors to two baths at different temperatures, it exhibits remarkable phenomena, including an unexpected 'freezing by heating. These phenomena are explored through systematic numerical simulations. This study reveals complicated relaxation processes as well as a crossover between two very different steady state regimes which are found to be bistable within a certain parameter range. / Ph. D.

In this thesis I study a model of self propelled particles exhibiting run-and tumble dynamics on lattice. This non-Brownian diffusion is characterised by a random walk with a finite persistence length between changes of direction, and is inspired by the motion of bacteria such as Escherichia coli. By defining a class of models with multiple species of particle and transmutation between species we can recreate such dynamics. These models admit exact analytical results whilst also forming a counterpart to previous continuum models of run-and- tumble dynamics. I solve the externally driven non-interacting and zero-range versions of the model exactly and utilise a field theoretic approach to derive the continuum fluctuating hydrodynamics for more general interactions. I make contact with prior approaches to run-and-tumble dynamics of lattice and determine the steady state and linear stability for a class of crowding interactions, where the jump rate decreases as density increases. In addition to its interest from the perspective of nonequilibrium statistical mechanics, this lattice model constitutes an efficient tool to simulate a class of interacting run-and-tumble models relevant to bacterial motion. Pattern formation in bacterial colonies is confirmed to be able to stem solely from the interplay between a diffusivity that depends on the local bacterial density and regulated division of the cells, in particular without the need for any explicit chemotaxis. This simple and generic mechanism thus provides a null hypothesis for pattern formation in bacterial colonies which has to be falsified before appealing to more elaborate alternatives. Most of the literature on bacterial motility relies on models with instantaneous tumbles. As I show, however, the finite tumble duration can play a major role in the patterning process. Finally a connection is made to some real experimental results and the population ecology of multiple species of bacteria competing for the same resources is considered.

We present predictions for the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms. Expressions for the fluid velocity, density and temperature are derived using equilibrium statistical mechanics. The results show that the number of samples needed to adequately resolve the flow-field scales as the inverse square of the Mach number. The theoretical results are verified for a dilute gas using direct Monte Carlo simulations. The agreement between theory and simulation verifies that the use of equilibrium theory is justified. / Singapore-MIT Alliance (SMA)

We investigate the properties of two distinct disordered systems: the two-species predator-prey Lotka-Volterra model with rate variability, and an elastic line model to simulate vortex lines in type-II superconductors. We study the effects of intrinsic demographic variability with inheritance in the reaction rates of the Lotka-Volterra model via zero-dimensional Monte Carlo simulations as well as two-dimensional lattice simulations. Individuals of each species are assigned inheritable predation efficiencies during their creation, leading to evolutionary dynamics and thus population-level optimization. We derive an effective subspecies mean-field theory and compare its results to our numerical data. Furthermore, we introduce environmental variability via quenched spatial reaction-rate randomness. We investigate the competing effects and relative importance of the two types of variability, and find that both lead to a remarkable enhancement of the species densities, while the aforementioned optimization effects are essentially neutral in the densities. Additionally, we collected extinction time histograms for small systems and find a marked increase in the stability of the populations against extinction due to the presence of variability. We employ an elastic line model to investigate the steady-state properties and non-equilibrium relaxation kinetics of magnetic vortex lines in disordered type-II superconductors. To this end, we developed a versatile and efficient Langevin molecular dynamics simulation code, allowing us to do a careful study of samples with or without vortex-vortex interactions or disorder allows us to disentangle the various complex relaxational features present in this system and investigate their origin. In particular, we compare disordered samples with randomly distributed point defects versus correlated columnar defects. We extract two-time quantities such as the mean-square displacement, the height and density correlations, to investigate the relaxation kinetics of the system of flux lines. Additionally, we compare the steady-state mean velocity and gyration radius as a function of an external driving current in the presence of point-like and columnar disorder. We validate our simulation algorithm by matching our results against a previously-used Monte Carlo algorithm, verifying that these microscopically quite distinct methods yield similar results even in out-of-equilibrium settings. / Ph. D.

We present predictions for the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms. Specifically, we present predictions for the error dependence on hydrodynamic parameters and the number of samples taken. Expressions for the common hydrodynamic variables of interest such as flow velocity, temperature, density, pressure, shear stress and heat flux are derived using equilibrium statistical mechanics. Both volume-averaged and surface-averaged quantities are considered. Comparisons between theory and computations using direct simulation Monte Carlo for dilute gases, and molecular dynamics for dense fluids, show that the use of equilibrium theory provides accurate results. / Singapore-MIT Alliance (SMA)

The thesis is divided in two parts, corresponding to the two main subjects on which I have worked during my PhD. In the first Part, we introduce many-body long-range interacting systems, such as plasma and self-gravitating systems. We first review the well known properties of isolated systems, which show peculiar behaviors both for what concern the equilibrium and the relaxation to equilibrium. We then consider long-range systems driven away from equilibrium and we show how the techniques developed for isolated systems can be extended to describe these situations. Generalizations to describe simplified models relevant for geophysical flows and two-dimensional turbulence are also discussed. Our work stands at the edge between the study of long-range interacting systems and the study of non-equilibrium systems.The second part of the thesis is devoted to the study of equilibrium properties of Hamiltonian systems with energy landscape techniques. A number of recent results is reviewed and applied to long and short-range interacting systems. One of the scope of my work was to study models whose energy landscape is much more complicated than what previously done. In the case of ferromagnetic short-range O(n) models on hypercubic lattices, our analysis unveiled a striking similarity between the critical energies of the Ising model and the O(n) models defined on the same lattice with the same interaction matrix. Generalizations of the Stillinger and Weber formalism are discussed as preliminary results and future perspectives.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19472号 / 理博第4132号 / 新制||理||1594(附属図書館) / 32508 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 佐々 真一, 准教授 藤 定義, 准教授 武末 真二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Natural ecologies are prone to stochastic effects and changing environments that shape their dynamical behavior. Ecological systems can be modeled through relatively simple population dynamics models. There is a plethora of models describing deterministic models of ecological systems evolving in a constant environment. However, stochasticity can lead to extinction or fixation events, noise-stabilized patterns, and nontrivial correlations. Likewise, changing environments can greatly affect the behavior and ultimate fate of ecological systems. In fact, the dynamics of evolution are mostly driven by randomness and changing environments. Therefore, it is of utmost importance to develop population dynamics models that are able to capture the effects of noise and environmental drive. In this thesis, we use both theoretical tools and simulations to investigate population dynamics in the following contexts: We study the stochastic spatial Lotka-Volterra (LV) model for predator-prey interaction subject to a periodically varying carrying capacity. The LV model with on-site lattice occupation restrictions that represent finite food resources for the prey exhibits a continuous active-to-absorbing phase transition. The active phase is sustained by spatio-temporal patterns in the form of pursuit and evasion waves. Monte Carlo simulations on a two-dimensional lattice are utilized to investigate the effect of seasonal variations of the environment on species coexistence. The results of our simulations are also compared to a mean-field analysis. We find that the parameter region of predator and prey coexistence is enlarged relative to the stationary situation when the carrying capacity varies periodically. The stationary regime of our periodically varying LV system shows qualitative agreement between the stochastic model and the mean-field approximation. However, under periodic carrying capacity switching environments, the mean-field rate equations predict period-doubling scenarios that are washed out by internal reaction noise in the stochastic lattice model. Utilizing visual representations of the lattice simulations and dynamical correlation functions, we study how the pursuit and evasion waves are affected by ensuing resonance effects. Correlation function measurements indicate a time delay in the response of the system to sudden changes in the environment. Resonance features are observed in our simulations that cause prolonged persistent spatial correlations. Different effective static environments are explored in the extreme limits of fast- and slow periodic switching. The analysis of the mean-field equations in the fast-switching regime enables a semi-quantitative description of the stationary state. The mean-field analysis of the Lotka-Volterra predator-prey model with seasonally varying carrying capacity is extended to the resonant regime. This is done by introducing a homotopy mapping from this model to another model that allows for the application of Floquet theory. The stability of the coexistence fixed point is studied and the period doubling is related to a bifurcation point in the homotopy mapping. However, we find that the predator-prey ecology's coexistence is stable for most of its parameter region. We apply a perturbative Doi–Peliti field-theoretical analysis to the stochastic spatially extended symmetric Rock-Paper-Scissors (RPS) and May–Leonard (ML) models, in which three species compete cyclically. Compared to the two-species Lotka–Volterra predator-prey (LV) model, according to numerical simulations, these cyclical models appear to be less affected by intrinsic stochastic fluctuations. Indeed, we demonstrate that the qualitative features of the ML model are insensitive to intrinsic reaction noise. In contrast, and although not yet observed in numerical simulations, we find that the RPS model acquires significant fluctuation-induced renormalizations in the perturbative regime, similar to the LV model. We also study the formation of spatio-temporal structures in the framework of stability analysis and provide a clearcut explanation for the absence of spatial patterns in the RPS system, whereas the spontaneous emergence of spatio-temporal structures features prominently in the LV and the ML models. Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reactiondiffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka–Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka–Volterra model as well as its May–Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse-graining in spatially extended stochastic reactiondiffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data. / Doctor of Philosophy / Population dynamics models describe how the number of individuals of interacting species changes over time. This is used to understand the ultimate fate of ecological systems. An ecological system can exhibit long-time multi-species coexistence, the fixation of just one species (all other species go extinct), or total extinction of the system. Understanding the dynamics of the system can help predict the final state of the system from early observations, also, it can inform possible ways to steer the system into a desirable outcome. However, it is very difficult to model such systems due to their complexity. While great progress has been made in understanding well-mixed populations in constant environments, there is still much to learn about ecological systems under spatial and environmental variability. A complete understanding of ecological dynamics and how they couple to evolutionary dynamics requires models of populations that are random, and that take into account how different species might be more or less dominant in different environments. We contribute to investigating these models in the following way: Seasonal variations in temperature leads to a change in the availability of different crops. This affects the resources available for animal species to consume in one season compared to another season (e.g. summer and winter). We study a predator-prey model wherein the resource abundance available to the prey vary between two seasons. We showcase how this affects the system's coexistence regime, and spatial patterns. Cyclic models of predation are models where the food chain is cyclic, meaning that there is no "food chain" but rather a "food circle". We utilize theoretical tools to gain a better understanding of the spontaneous formation of well-known spatial patterns in cyclic predation models. The aim of population dynamics is to write a simple set of equations or models that can accurately capture the behavior of natural ecologies. This is rarely an easy task, because even if the microscopic interactions between species is known, it is very difficult to simplify this microscopic model to a simple set of macroscopic equations. We develop a technique that uses computer simulations to map microscopic interactions into simple rate equations. This work can inform better modeling of observational data.

Active particle is a general term used to label a large set of different systems, spanning from a flock of birds flying in a coordinated pattern to a school of fish abruptly changing its direction or to a bacterium self-propelling itself while foraging nourishment. The common property shared by these systems is that their constituent agents, e.g. birds, fishes, or bacteria, are capable of harvesting energy from the surrounding environment and converting it into self-propulsion and directed motion. This peculiar feature characterizes them as out-of-equilibrium systems, in fact, the process of energy consumption and dissipation generates microscopically irreversible dynamics and drives them far from thermal equilibrium. Thanks to their intrinsic out-of-equilibrium nature, active particle systems often display characteristic patterns and behaviors that are not observed in equilibrium physics systems, such as collective motion or motility-induced phase separation. These features prompted the development of theories and algorithms to simulate and study active particles, giving rise to paradigmatic models capable of describing these phenomena, such as the Vicsek model for collective motion, the run-and-tumble model, or the active Brownian particle model. At the same time, synthetic agents have been designed to reproduce the behaviors of these natural active particle systems, and their evolution could play a fundamental role in the nanotechnology of the 21st century and the development of novel medical treatments, in particular controlled drug delivery. A specific type of active particle that uses its directed motion to move at the microscale is called a microswimmer. Examples of these agents are bacteria exploring their surroundings while searching for food or escaping external threats, spermatozoa looking for the egg, or artificial Janus particles designed for specific tasks. Active agents at these scales use different swimming mechanisms, such as rotating flagella or phoretic motion along chemical gradients that they can create. The outcome of their efforts is determined by the interplay of the translational diffusion intrinsic to the dynamics at these scales and the persistent motion that characterizes their self-propulsion. The problem of finding a specific target in a complex environment is essential for microswimmers and active agents in general. Target search is employed by animals and microorganisms for a variety of purposes, from foraging nourishment to escaping potential threats, such as in the case of bacterial chemotaxis. The study of this process is therefore fundamental to characterize the behavior of these systems in nature. Its complete description could then be employed in designing synthetic microswimmers for addressing specific problems, such as the aforementioned targeted drug delivery and the environmental cleansing of soil and polluted water. Here, we provide a detailed study of the target search process for microswimmers exploring complex environments. To this end, we generalize Transition Path Theory, the rigorous statistical mechanics description of transition processes, to the target-search problem. The most general way of modeling a complex environment that the microswimmer has to navigate is through an external potential. This potential can be characterized by high barriers separating metastable states in the system or by the presence of confining boundaries. If a high energy barrier is located between the initial position of the microswimmer and its target, the target search becomes a rare event. Rare events have been thoroughly investigated in equilibrium physics, and several algorithms have been designed to cope with the separation of timescales intrinsic to these problems and enable their investigation via efficient computer simulations. Despite the large set of tools developed for studying passive particles performing rare transitions, the characterization of this process for non-equilibrium systems, such as active particles, is still lacking. One of the main results of this thesis is the generalization to non-equilibrium systems of the Transition Path Sampling (TPS) algorithm, which was originally designed to simulate rare transitions in passive systems. This algorithm relies on the generation of productive trajectories, i.e. trajectories linking the initial state of the particle to the target state, via a Monte Carlo procedure, without the need of simulating long thermal oscillations in metastable states. These trajectories are then accepted according to a Metropolis criterion and are subsequently used to obtain the transition path ensemble, i.e. the ensemble of all reactive paths that completely characterizes the process. The TPS algorithm relies on microscopic reversibility to generate the productive trajectories, therefore its generalization to out-of-equilibrium systems lacking detailed balance and microscopic reversibility has remained a major challenge. Within this work, after deriving a path integral representation for active Brownian particles, we provide a new rule for the generation and acceptance of productive non-equilibrium trajectories, which reduces to the usual expression for passive particles when the activity of the microswimmer is set to zero. This new rule allows us to generalize the TPS algorithm to the case of active Brownian particles and to obtain a first insight into the counterintuitive target-search pathways explored by these particles. In fact, while passive particles perform barrier crossing following the minimum energy path linking the initial state to the target state, we found that active particles, thanks to their activity and persistence of motion, can reach the target more often by surfing higher energy regions of the landscape that lie far from the minimum energy path. The second result of this thesis is a systematic characterization of the target-search path ensemble for an active particle exploring an energy landscape. We do so by analyzing the system’s response to changes in the two adimensional parameters that define the parameter space of the model: the Péclet number and the persistence of the active particle. Our findings show that active Brownian particles can increase their target-finding rates by tuning their Péclet number and their persistence according to the shape and characteristics of the external landscape. We perform this analysis in two different landscapes, namely a double-well potential and the Brown-Müller potential, finding robust features in the target-search patterns. In contrast, other observables of the system, e.g. the target-finding rates, are more responsive to the features of the external environment. Interestingly, our results suggest that, differently from what happens for passive particles, the presence of additional metastable states in the system does not hinder the target-search dynamics of active particles. The third original contribution of this Ph.D. thesis is the generalization of the concept of the committor function to target-search problems. The committor function was first introduced in the framework of Transition Path Theory to study reaction processes. If a definition for a reactant and a product state embedded in the configuration space of the system is provided, the committor function quantifies the probability that a trajectory starting in a given configuration reaches the product state before it can enter the reactant. For this reason, it has been proven to be pivotal for a complete characterization of these events and it is often regarded as the optimal reaction coordinate for thermally activated transitions. The target search problem shares many similarities with transition processes since it is characterized by an initial state from which the agent begins its journey and a target state that the particle is aiming to reach, and often some barriers or obstacles separate the two. Exploiting these similarities, we take advantage of the concept of the committor function to fully characterize a target-search process performed by an active agent. First, we derive the Fokker-Planck equation for an active Brownian particle subject to an external potential, and we use its associated probability current to define the committor function for an active agent. Then, we prove that the active committor satisfies the Backward-Kolmogorov equation analogously to the committor for passive particles. We take advantage of this property to efficiently compute the committor function using a finite-difference algorithm, validating it with brute-force simulations. Finally, we further validate our theory with experiments of a camphor self-propelled disk. This self-propelled disk is capable of moving on a water surface and is studied during its exploration of a circular confining environment. We start by analyzing long recorded trajectories of such a disk moving in a Petri dish, and, after defining a reactant and a product region in the system, we proceed to compute the committor function in three different regions contained in the dish. We analyze all the trajectory slices passing through those regions and we measure how many of them hit the product region and how many hit instead the reactant first, and we obtain the committor in the three regions as a function of the angle. Finally, we simulate a long trajectory of an active Brownian particle exploring a circular confining environment, and we compare the committor as an angular function obtained from brute-force simulations with the committor estimated from experimental data.