No description available.

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

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.

Consideramos dois tópicos distintos relacionados a modelos clássicos da mecânica estatísticas de equilíbrio. O primeiro constitui-se na análise de equação parabólicas semi-lineares associadas à transformação de grupo de renormalização para o gás de Coulomb hierárquico bidimensional e o gás dipolos hierárquicos em dimensão d>1 após tomarmos um limite apropriado (limite L 1 do tamanho do bloco). O outro tópico estudado foi a construção de uma função majorante (, z) para a pressão termodinâmica de um gás formado por partículas interagentes com atividade z e temperatura -1, cuja interação entre dois corpos pode ser decomposta em escalas como um potencial estável. Somos capazes de demonstrar que o problema de valor inicial dado pela equação do gás de Coulomb está bem definido (existência, unicidade e dependência contínua das soluções) em um espaço funcional adequado e a solução converge assintoticamente para uma das infinitas contáveis soluções de equilíbrio. Quanto ao gás de dipolos, embora não tenhamos conseguido provar a existência e unicidade das soluções, garantimos que a única solução estacionária limitada inferiormente é a trivial nula, que é uma solução estável. Ao menos no caso dos modelos hierárquicos, os resultados obtidos permitem dar uma resposta definitiva à conjectura de Gallavotti e Nicolò sobre uma sequência infinita de transições de fase. A função majorante é construída como a solução de uma equação diferencial parcial quase-linear de primeira ordem. Através da do método das características relacionamos a solução (majorante) à função W de Lambert cuja expansão em série possui uma singularidade originada pelo corte que a função W possui no plano complexo. A descrição da função majorante como uma função W possui no plano complexo. A descrição da função majorante como uma função W permite uma melhora nas estimativas de raio de convergência para série de Mayer para pressão. / We have considered in this thesis two distinct topics related to classic models in equilibrium statistical mechanics. The first one is the analysis of semilinear parabolic partial differential equations given by a suitable limit (size of block L 1) in the renormalization group for the dipole gas in any dimension d>1. The other topic is the construction of a majorant function (, z) for the thermodynamic -1 whose potential admits a scale decomposition in terms of some stable potential. We are capable to demonstrate the well-posedness (existence, uniqueness and continuous dependence of solutions) for Coulomb gas equations and the global asymptotic convergence of the flow to one of its countably many equilibrium solutions. The dipole gas equations are technically more difficult and lack the results weve achieved in Coulomb gas but, despite its difficulties, we can establish the uniqueness of the trivial solution as a equilibrium ane and its stabilish. At least for hierarchical models, the established results give a definite answer to Gallovotti and Niclolòs conjecture of na infinite of phase transitions. The majorant function is constructed as the solution of a first order quase-linear partial differential equation. By means of the characteristics method we are able to relate its solution (the majorant) to Lamberts W-function whose series expansion possess a singularity given by W-function allows better estimates for Mayer series convergence.