The transportation of mass is an inherently `non-equilibrium' process, relying on a current of mass between two or more locations. Life exists by necessity out of equilibrium and non-equilibrium transport processes are seen at all levels in living organisms, from DNA replication up to animal foraging. As such, biological processes are ideal candidates for modelling using non-equilibrium stochastic processes, but, unlike with equilibrium processes, there is as of yet no general framework for their analysis. In the absence of such a framework we must study specific models to learn more about the behaviours and bulk properties of systems that are out of equilibrium. In this work I present the analysis of three distinct models of non-equilibrium mass transport processes. Each transport process is conceptually distinct but all share close connections with each other through a set of fundamental nonequilibrium models, which are outlined in Chapter 2. In this thesis I endeavour to understand at a more fundamental level the role of stochasticity and fluctuations in non-equilibrium transport processes. In Chapter 3 I present a model of a diffusive search process with stochastic resetting of the searcher's position, and discuss the effects of an imperfection in the interaction between the searcher and its target. Diffusive search process are particularly relevant to the behaviour of searching proteins on strands of DNA, as well as more diverse applications such as animal foraging and computational search algorithms. The focus of this study was to calculate analytically the effects of the imperfection on the survival probability and the mean time to absorption at the target of the diffusive searcher. I find that the survival probability of the searcher decreases exponentially with time, with a decay constant which increases as the imperfection in the interaction decreases. This study also revealed the importance of the ratio of two length scales to the search process: the characteristic displacement of the searcher due to diffusion between reset events, and an effective attenuation depth related to the imperfection of the target. The second model, presented in Chapter 4, is a spatially discrete mass transport model of the same type as the well-known Zero-Range Process (ZRP). This model predicts a phase transition into a state where there is a macroscopically occupied `condensate' site. This condensate is static in the system, maintained by the balance of current of mass into and out of it. However in many physical contexts, such as traffic jams, gravitational clustering and droplet formation, the condensate is seen to be mobile rather than static. In this study I present a zero-range model which exhibits a moving condensate phase and analyse it's mechanism of formation. I find that, for certain parameter values in the mass `hopping' rate effectively all of the mass forms a single site condensate which propagates through the system followed closely by a short tail of small masses. This short tail is found to be crucial for maintaining the condensate, preventing it from falling apart. Finally, in Chapter 5, I present a model of an interface growing against an opposing, diffusive membrane. In lamellipodia in cells, the ratcheting effect of a growing interface of actin filaments against a membrane, which undergoes some thermal motion, allows the cell to extrude protrusions and move along a surface. The interface grows by way of polymerisation of actin monomers onto actin filaments which make up the structure that supports the interface. I model the growth of this interface by the stochastic polymerisation of monomers using a Kardar-Parisi-Zhang (KPZ) class interface against an obstructing wall that also performs a random walk. I find three phases in the dynamics of the membrane and interface as the bias in the membrane diffusion is varied from towards the interface to away from the interface. In the smooth phase, the interface is tightly bound to the wall and pushes it along at a velocity dependent on the membrane bias. In the rough phase the interface reaches its maximal growth velocity and pushes the membrane at this speed, independently of the membrane bias. The interface is rough, bound to the membrane at a subextensive number of contact points. Finally, in the unbound phase the membrane travels fast enough away from the interface for the two to become uncoupled, and the interface grows as a free KPZ interface. In all of these models stochasticity and fluctuations in the properties of the systems studied play important roles in the behaviours observed. We see modified search times, strong condensation and a dramatic change in interfacial properties, all of which are the consequence of just small modifications to the processes involved.

For systems that are in equilibrium, fluctuations can be understood through interactions with external heat reservoirs. For this reason these fluctuations are known as thermal noise, and they usually become vanishingly small in the thermodynamic limit. However, many systems comprising interacting constituents studied by physicists in recent years are both far from equilibrium, and sufficiently small so that they must be considered finite. The finite number of constituents gives rise to an inherent demographic noise in the system, a source of fluctuations that is always present in the stochastic dynamics. This thesis investigates the role of stochastic fluctuations in the macroscopically observable dynamical behaviour of non-equilibrium, finite systems. To facilitate such a study, we construct microscopic models using an individual based modelling approach, allowing the explicit form of the demographic noise to be identified. In many physical systems and theoretical models, absorbing states are a defining feature. Once a system enters one, it cannot leave. We study the dynamics of a system with two symmetric absorbing states, finding that the amplitude of the multiplicative noise can induce a transition between two universal modes of domain coarsening as the system evolves to one of the absorbing states. In biological and ecological systems, cycles are a ubiquitously observed phenomenon, but are di cult to predict analytically from stochastic models. We examine a potential mechanism for cycling behaviour due to the flow of probability currents, induced by the athermal nature of the demographic noise, in a single patch population comprising two competing species. We find that such a current by itself cannot generate macroscopic cycles, but when combined with deterministic dynamics which constrain the system to a closed circular manifold, gives rise to global quasicycles in the population densities. Finally, we examine a spatially extended system comprising many such patch populations, exploring the emergence of synchronisation between the different cycles. By a stability analysis of the global synchronised state, we probe the relationship between the synchronicity of the metapopulation and the magnitude of the coupling between patches due to species migration. In all cases, we conclude that the nature of the demographic noise can play a pivotal role in the macroscopically observed dynamical behaviour of the system.

The study of non-equilibrium physics is an area of interest since, unlike for their equilibrium counterparts, there exists no general framework for solving such systems. In this thesis I investigate the emergence of structure and front propagation in driven systems, a special type of system within the area of non-equilibrium physics. In particular I focus on three particular one-dimensional models each of which illustrate this in a different way. The Driven Asymmetric Contact Process (DACP) describes a system where activity is continuously generated at one end of a one-dimensional lattice and where this activity is allowed to spread in one direction along the lattice. In the DACP one observes a propagating wave of activity which appears to abruptly vanish as the system undergoes a phase transition. Using a modified Fisher equation to model the system reveals the continued existence of the propagating wave, now contained within a decaying envelope. Furthermore this establishes relations between properties of the travelling wave and Directed Percolation critical exponents. The Zero-Range Process (ZRP) is a much studied system exhibiting a condensation transition. In the ZRP individual particles hop along a lattice at rates which depend only on the occupancy of the departure site. Here I investigate a modi cation of the ZRP where instead the majority of the particles at a site depart during a single hopping event. For this, the Chipping model, a condensate which propagates along the lattice is observed. It is found that this condensation transition is present even for hop rates which fall foul of the condensation requirements of the normal ZRP. Further it is observed that, unlike for normal ZRP, condensation occurs even in the low-density limit. As a result I suggest a condensation mechanism which depends only on the hop rates of low occupancy sites. The Host-Solute-Vacancy model (HSV) is a three-species system designed to model electromigration in a circuit. As the parameter space is navigated the system undergoes what appear to be two separate phase transitions from a randomly distributed state to a condensed state with either of two structures. To investigate the model new measures for determining condensation are developed. These show that, again, condensation occurs in the low-density limit. By a reduction to a ZRP an effective hop rate of the system is measured. This effective hop rate is found to beta function of the occupancy of a site as a fraction of the total system size. To explain this behaviour I invoke a description whereby there is a step in the hop rate as a function of occupancy. Through these three examples I illustrate how minor modi cations to the dynamics of known systems can result in a new and rich phenomenology. I draw particular attention to the effect of asymmetry in the dynamics.

We study game theoretic ecological models with cyclic competition in the case where the strategies can be mixed or pure. For both projects, reported in [49] and [50], we employ Monte Carlo simulations to study finite systems. In chapter 3 the results of a previously published paper [49] are presented and expanded upon, where we study the extinction time of four cyclically competing species on different lattice structures using Lotka-Volterra dynamics. We find that the extinction time of a well mixed system goes linearly with respect to the system size and that the probability distribution approximately takes the shape of a shifted exponential. However, this is not true for when spatial structure is added to the model. In that case we find that instead the probability distribution takes on a non-trivial shape with two characteristic slopes and that the mean goes as a power law with an exponent greater than one. This is attributed to neutral species pairs, species who do not interact, forming domains and coarsening. In chapter 4 the results of [50] are reported and expanded, where we allow agents to choose cyclically competing strategies out of a distribution. We first study the case of three strategies and find through both simulation and mean field equations that the probability distributions of the agents synchronize and oscillate with time in the limit where the agents probability distributions can be approximated as continuous. However, when we simulate the system on a one-dimensional lattice and the probability distributions are small and discretized, it is found that there is a drastic transition in stability, where the average extinction time of a strategy goes from being a power law with respect to system size to an exponential. This transition can also be observed in space time images with the emergence of tile patterns. We also look into the case of four cyclically competing strategies and find results similar to that of [49], such as the coarsening of neutral domains. However, the transition from power law to exponential for the average extinction time seen for three strategies is not observed, but we do find a transition from one power law to another with a different slope. This work was supported by the United States National Science Foundation through grants DMR-0904999 and DMR-1205309. / Ph. D.

All real-world populations are composed of a finite number of individuals. Due to the intrinsically random nature of interactions between individuals, the dynamics of finite-sized populations are stochastic processes. Additionally, for many types of interaction not all effects occur instantaneously. Instead there are delays before effects are felt. The centrepiece of this thesis is a method of analytically studying stochastic population dynamics with delay reactions. Dynamics with delay reactions are non-Markovian, meaning many of the widely used techniques to study stochastic processes break down. It is not always possible to formulate the master equation, which is a common starting point for analysis of stochastic effects in population dynamics. We follow an alternative method, and derive an exact functional integral approach which is capable of capturing the effects of both stochasticity and delay in the same modelling framework. Our work builds on previous techniques developed in statistical physics, in particular the Martin-Siggia-Rose-Janssen-de Dominicis functional integral. The functional integral approach does not rely on an particular constraints on the population dynamics, for example the choice of delay distribution. Functional integrals can not in general be solved exactly. We show how the functional integral can be used to derive the deterministic, chemical Langevin, and linear-noise approximations for stochastic dynamics with delay. In the later chapters we extend Gillespie’s approximate method of studying stochastic dynamics with delay reactions, which can be used to derive the chemical Langevin equation, by-pass the functional integral. We also derive an extension to the functional integral approach so that it also covers systems with interruptible delay reactions. To demonstrate the applicability of our results we consider various models of population dynamics, arising from ecology, epidemiology, developmental biology, and chemistry. Our analytical calculations are found to provide excellent agreement with exact numerical simulations.

Recent years have seen remarkable progress in our understanding of physical aging in nondisordered systems with slow, i.e. glassy-like dynamics. In many systems a single dynamical length L(t), that grows as a power-law of time t or, in much more complicated cases, as a logarithmic function of t, governs the dynamics out of equilibrium. In the aging or dynamical scaling regime, these systems are best characterized by two-times quantities, like dynamical correlation and response functions, that transform in a specific way under a dynamical scale transformation. The resulting dynamical scaling functions and the associated non-equilibrium exponents are often found to be universal and to depend only on some global features of the system under investigation. We discuss three different types of systems with simple and complex aging properties, namely reaction diffusion systems with a power growth law, driven diffusive systems with a logarithmic growth law, and a non-equilibrium polymer network that is supposed to capture important properties of the cytoskeleton of living cells. For the reaction diffusion systems, our study focuses on systems with reversible reaction diffusion and we study two-times functions in systems with power law growth. For the driven diffusive systems, we focus on the ABC model and a related domain model and measure two- times quantities in systems undergoing logarithmic growth. For the polymer network model, we explain in some detail its relationship with the cytoskeleton, an organelle that is responsible for the shape and locomotion of cells. Our study of this system sheds new light on the non- equilibrium relaxation properties of the cytoskeleton by investigating through a power law growth of a coarse grained length in our system. / Ph. D.

We numerically investigate the dynamical properties of driven magnetic flux vortices in disordered type-II superconductors for a variety of temperatures, types of disorder and sample thicknesses. We do so with the aid of Langevin molecular dynamics simulations of a coarsegrained elastic line model of flux vortices in the extreme London limit. Some original findings of this doctoral work include the discovery that flux vortices driven through random point disorder show simple aging following drive quenches from the moving lattice state to both the pinned glassy state (non-universal aging) and near the critical depinning region (universal aging); estimations of experimentally consistent critical scaling exponents for the continuous depinning phase transition of vortices in three dimensions; and an estimation of the boundary curve separating regions of linear and non-linear electrical transport for flux lines driven through planar defects via novel direct measurements of vortex excitations. / Ph. D. / The works contained in this dissertation were undertaken with the goal of better understanding the dynamics of driven magnetic flux lines in type-II superconductors under different conditions of temperature, material defects and sample thickness. The investigations were conducted with the aid of computer simulations of the flux lines which preserve physical aspects of the system relevant to long-time dynamics while discarding irrelevant microscopic details. As a result of this work, we found (among other things) that when driven by electric currents, flux lines display very different dynamics depending on the strength of the current. When the current is weak, the material defects strongly pin the flux lines leaving them in a disordered glassy state. Sufficiently high current overpowers the defect pinning and results in the flux lines forming into a highly ordered crystal-like structure. In the intermediate critical current regime, the competing forces become comparable resulting in very large fluctuations of the flux lines and a critical slowing down of the flux line dynamics.

We investigate the relaxation properties of two distinct systems: magnetic vortex lines in disordered type-II superconductors and charge carriers in the Coulomb glass in disordered semiconductors. We utilize an elastic line model to simulate magnetic flux lines in disordered type-II superconductors by performing Langevin molecular dynamics simulations. We study the non-equilibrium relaxation properties of flux lines in the presence of uncorrelated point-like disorder or extended linear defects analyzing the effects of rapid changes in the system's temperature or magnetic field on these properties. In a previously-equilibrated system, either the temperature is suddenly changed or the magnetic field is abruptly altered by adding or removing random flux lines to or from the system. One-time observables such as the radius of gyration are measured to characterize steady-state properties, and two-time correlation functions such as the vortex line height autocorrelations are computed to investigate the relaxation dynamics in the aging regime and therefore distinguish the complex relaxation features that result from the different types of disorder in the system. This study allows us to test the sensitivity of the system's non-equilibrium aging kinetics to the selection of initial states and to make closer contact to experimental setups. Furthermore, we employ Monte Carlo simulations to study the relaxation properties of the two-dimensional Coulomb glass in disordered semiconductors and the two-dimensional Bose glass in type-II superconductors in the presence of extended linear defects. We investigate the effects of adding non-zero random on-site energies from different distributions on the properties of the correlation-induced Coulomb gap in the density of states and on the non-equilibrium aging kinetics highlighted by the autocorrelation functions. We also probe the sensitivity of the system's equilibrium and non-equilibrium relaxation properties to instantaneous changes in the density of charge carriers in the Coulomb glass or flux lines in the Bose glass. / Ph. D.

Cette thèse porte sur la physique statistique des systèmes hors d’équilibre maintenus dans un état stationnaire. Plus spécifiquement, ce travail s’intéresse à des quantités macroscopiques conservées (le volume, la masse, etc.) qui peuvent être échangées entre plusieurs systèmes hors d’équilibre en contact. Cette mise en contact d’un ou plusieurs systèmes est une situation fondamentale en thermodynamique classique des systèmes à l’équilibre, en ce qu’elle permet de définir la notion de paramètre thermodynamique conjugué comme la température, la pression, le potentiel chimique, etc., qui dérivent d’un même potentiel thermodynamique. Dans les systèmes hors d’équilibre stationnaires, l’existence de tels paramètres conjugués dérivant d’un potentiel thermodynamique (énergie libre) demeure une question ouverte. En se focalisant sur la situation du contact entre deux systèmes stochastiques hors d’équilibre quelconques de particules sur réseau dans des états homogènes, nous montrons l’existence d’une fonction de grande déviation attachée aux densités globales des deux systèmes, lorsque la fréquence d’échange de particules entre ces derniers est faible. Cette fonction de grandes déviations hors d'équilibre, analogue de l’énergie libre, vérifie une équation dite de Hamilton-Jacobi. Nous identifions les conditions naturelles pour lesquelles la fonction de grandes déviations est additive, menant ainsi à la définition de potentiels chimiques hors-équilibre. Néanmoins, nous montrons que ceux-ci dépendent de façon générique de la dynamique au contact et ne vérifient donc pas d’équation d’état. En l’absence de bilan détaillé macroscopique, l’équation de Hamilton-Jacobi est beaucoup plus difficile à résoudre. Une analyse perturbative par rapport aux forçages hors-équilibres permet de se convaincre que l’additivité est génériquement brisée dès les premiers ordres de perturbation en l’absence de bilan détaillé. Au-delà de la propriété d’additivité, cette fonction de grandes déviations peut être liée dans un certain nombre de cas au travail exercé par un potentiel extérieur à travers une relation de type second principe de la thermodynamique. Nous discutons également différentes façons d’y avoir accès expérimentalement.Fort de cette analyse théorique générale, nous illustrons celle-ci sur des systèmes stochastiques sur réseau classiques (Zero Range Process et Driven Lattice Gases) ainsi que sur un modèle de transport de masse original, exactement soluble. Nous appliquons également notre analyse sur des systèmes de particules auto-propulsées indépendantes. Dans chaque cas, l’importance du contact est alors pleinement révélée, en accord avec la littérature récente, que ce soit au niveau de la dynamique elle-même ou de la position de ce dernier vis-à-vis des systèmes. / This thesis deals with the statistical physics of out-of-equilibrium systems maintained in a steady state. More specifically, this work focuses on macroscopic conserved quantities (volume, mass, etc.) that can be exchanged between several out-of-equilibrium systems brought into contact. The contact between two systems is a fundamental situation in classical thermodynamics of equilibrium systems, since it allows one to define the notion of intensive thermodynamic parameter such as temperature, pressure, chemical potential, etc., derived from the same thermodynamic potential. For non-equilibrium steady state systems, the general existence of such intensive parameters remains an open issue. By focusing on the contact situation between two out-of-equilibrium stochastic systems on lattice in homogeneous states, we show the existence of a large deviation function attached to the overall densities of both systems, when the frequency of particle exchange between them is low. This large deviations function, analogous to a free energy, satisfies a so-called Hamilton-Jacobi equation. We identify the natural conditions for which the large deviation function is additive, leading to the definition of non-equilibrium chemical potentials. Nevertheless, we show that the latter generically depends on the contact dynamics and therefore do not obey any equation of state. In the absence of a macroscopic detailed balance, the Hamilton-Jacobi equation is much more difficult to solve. A perturbative analysis with respect to the driving forces allows one to show that additivity is generically broken. Beyond this additivity property, this large deviations function can – under certain assumptions – be related to the work applied by an external potential through a generalisation of the second law. We also discuss different ways to get access experimentally to this out-of-equilibrium free energy.Based on this general theoretical analysis, we eventually provide several illustrations on standard stochastic lattice models (Zero Range Process and Driven Lattice gases in particular) as well as a detailed analysis of an original, exactly solvable, mass transport model. Standard models of independent self-propelled particles are also discussed. The importance of the contact is eventually fully revealed, in agreement with recent literature, either in terms of the dynamics at contact itself or because of its position with respect to both systems.

[pt] Organismos vivos têm seus próprios meios de locomoção e são capazes de se reproduzir. Além disto, o habitat no qual os organismos estão inseridos é tipicamente heterogêneo, de modo que as condições ambientais variam no tempo e no espaço. Nesta tese, são propostos e investigados modelos teóricos para compreender o comportamento coletivo de organismos vivos, visando responder questões relevantes sobre a organização e preservação da população utilizando técnicas analíticas e numéricas. Inicialmente, considerando um habitat homogêneo, em que as propriedades estatísticas das condições ambientais são independentes do tempo e do espaço, estudamos como padrões espaço-temporais podem emergir na distribuição da população devido a interações não-locais e investigamos o papel das flutuações ambientais neste processo. Em seguida, assumindo um meio ambiente heterogêneo, analisamos o caso de um único domínio de habitat. Considerando uma classe de equações não lineares, introduzindo flutuações temporais e interações entre os organismos, fornecemos uma perspectiva geral da estabilidade de populações neste caso, desafiando os conceitos ecológicos anteriores. Em um segundo passo, assumindo uma paisagem complexa fragmentada, consideramos que os indivíduos têm acesso a informações sobre a estrutura espacial do meio. Mostramos que os indivíduos sobrevivem quando as regiões espaciais viáveis estão suficientemente aglomeradas e observamos que o tamanho da população é maximizado quando os indivíduos utilizam parcialmente a informação do meio ambiente. Finalmente, como resultados exatos analíticos não são factíveis em muitas situações importantes, propomos uma abordagem efetiva para interpretar os dados experimentais. Assim, somos capazes de conectar a heterogeneidade do ambiente e a persistência da população, caracterizada pela distribuição de probabilidade para os tempos de vida. / [en] Living entities have their own means of locomotion and are capable of reproduction. Furthermore, the habitat in which organisms are embedded is typically heterogeneous, such that environment conditions vary in time and space. In this thesis, theoretical models to understand the collective dynamics of living beings have been proposed and investigated aiming to address relevant questions such as population organization and persistence in the environment, using analytical and numerical techniques. Initially, considering an homogeneous habitat, in which the statistical properties of the environmental conditions are time and space independent, we study how spatiotemporal order can emerge in the population distribution due to nonlocal interactions and investigate the role of environment fluctuations in the self-organization process. Further, we continue our investigation assuming an heterogeneous environment, starting with the simplest case of a single habitat domain, and we obtain the critical conditions for population survival for different population dynamics. Considering a class of nonlinear equations, introducing temporal oscillations and interactions among the organisms, we are able to provide a general picture of population stability in a single habitat domain, challenging previous ecological concepts. At last, assuming a fragmented complex landscape, resembling realistic properties observed in nature, we additionally assume that individuals have access to information about the spatial structure. We show that individuals survive when patches of viable regions are clustered enough and, counter-intuitively, observe that population size is maximized when individuals have partial information about the habitat. Finally, since, analytical exact results are not feasible in many important situations, we propose an effective approach to interpret experimental data. This way we are able to connect environment heterogeneity and population persistence.