This thesis is mainly concerned with the problem of exponential convergence to equilibrium for open classical systems. We consider a model of a small Hamiltonian system coupled to a heat reservoir, which is described by the Generalized Langevin Equation (GLE) and we focus on a class of Markovian approximations to the GLE. The generator of these Markovian dynamics is an hypoelliptic non-selfadjoint operator. We look at the problem of exponential convergence to equilibrium by using and comparing three different approaches: classic ergodic theory, hypocoercivity theory and semiclassical analysis (singular space theory). In particular, we describe a technique to easily determine the spectrum of quadratic hypoelliptic operators (which are in general non-selfadjoint) and hence obtain the exact rate of convergence to equilibrium.

In this study, a novel lattice Boltzmann model (LBM) of CO2 dissolution at porous scale is proposed and developed to predict the CO2 dispersion and dissolution in geo-formations. The developed LBM dissolution model consists of an interfacial momentum interaction model, a mass transfer model and a convection (advection) model. Shen-Chen’s pseudopotential model using Equation of State (EOS) of real fluids is tested for momentum interaction model. It is found that a sharp interface can be maintained by optimizing the interaction strengths of two fluids with minimum numerical diffusion in the interfacial momentum interaction model. This makes it possible to model physical diffusion and interfacial tension individually. A new diffusion force, describing the particle diffusion driving by chemical potential at given solubility, is proposed for mass transfer model by applying the interparticle interaction pseudopotential concept. The dissolution is governed by coupling mechanism of diffusion and convection. The interface between the solute of CO2 and solvent water is monitored by the solubility, which changes and indicates the moving of interface as CO2 dissolving. The solution is considered as the mixture of dissolved CO2 and water. Instead of using an additional Lattice that is requested by the existed LBM, the further dispersion of dissolved solutes is attached to the Lattice of water, by which the cost of computing memory size and time is significantly reduced. The developed LBM dissolution model is calibrated by the data from Lab experiment of dissolution of CO2 droplet in water at a state of CO2 geological storage about 1000m depth. The calibration is made by comparison of simulation results with the data, in terms of the shrinking rate of CO2 droplet and the concentration distribution of dissolved CO2 in the solution layer. As the whole, the numerical predictions are well agreement with those of lab experiment. The developed model is then applied to investigate the mechanism of dispersion and dissolution of CO2 droplet in channels at pore scale, in terms of the effects of the Eo number, channel width and channel tilt angle. It is found under the state at 1000m depth that it is difficult for a dissolving CO2 droplet, unlike that of an immiscible droplet, to reach to a ’terminal velocity’. Because of the shrinking, dissolving CO2 droplets accelerate from a quiescent state to a maximum velocity and then decelerate in the channels. The ratio of droplet diameter (Do) to channel width (Lx), M=Do/Lx, and the inclination are the parameters that significantly affect the dynamics of dissolving CO2 droplets. The smaller the channel width or the tilt angle of the pores of the geoformation, the slower of stored CO2 can penetrate vertically and dissolve out. While, as the channel width increases to provide enough space, M<1, the shrinking rate is independent of the channel width and wobbling of droplets is observed at the region with the Re number of 300-600 and the Eo number of 20-43. The interactions of droplets in the channels (M=1 and M=0.3) are investigated by simulating of a pair of droplets dispersion and dissolution, with an initial distance of 4.5 times of droplet diameter. Comparison is made to that of single droplet in terms of the rising velocity and shrinking rate. It is found that the shrinking rate of the upper droplet is larger than that of the following droplet when the following droplet moves into the solution field of the upper droplet. The following droplet rises, when M=1 and M=0.3, faster than that of the upper droplet and also than that of the single droplet under the same conditions. The coalescence of two droplets is observed in the channel at M=0.3, which is due to the action of tail vortex of the upper droplet on the following droplet. The following droplet accelerates at a different wobbling frequency with that of the upper droplet. As the implication in model development, in term of numerical stability, the so called ’non-linear implicit trapezoidal lattice Boltzmann scheme’, proposed by Nourgaliev et al. [1], is re-examined in order to simulate the large density ratio of two-fluid flows. It is found from the re-derivation that the scheme is a linear scheme in nature. Therefore, the re-derived scheme is more efficient and the CPU time can be reduced. The test cases of the simulation of a steady state droplet using SC EOS show that re-derived scheme improves the numerical stability by reducing the spurious velocity about 21.7% and extending the density ratio 53.4% as relaxation time of the improved scheme is 0.25, in comparison to those from the traditional explicit scheme. Meanwhile, in the multicomponent simulation, with the same density distribution at steady state, the improved scheme reduces both the magnitude and spreading region of the spurious velocity. The spurious velocity of the improved method reduces approximate 4 times than that of the explicit scheme.

This thesis is based on the application of renormalization group techniques to examine a variety of characteristics of quasiperiodically forced systems. The initial focus of the work is symmetric barrier billiards, a pseudo-integrable system consisting of a particle moving at constant speed in a rectangular chamber with a partial barrier placed in the centre. A renormalization analysis of the autocorrelation function (ACF) is presented for a class of quadratic irrational trajectories, and depending on the nature of the barrier, this can lead to either self-similar or chaotic behaviour of the correlations. In the case of the golden mean trajectory, this is then explained by constructing a map which identifies the action of the renormalization operator with a subshift of finite type, and it is shown that orbits of the renormalization operator in a space of pairs of piecewise constant functions explore a specified attracting set. A projection of the function pairs in this set obtained by averaging them (to yield the correlations) gives rise to the presence of invariant sets embedded in three-dimensional space on which the correlations lie. We extend this work by giving a renormalization analysis of correlations in a quasiperiodically forced two-level quantum system in a time dependent magnetic field, which consists of periodic kicks whose amplitude is determined by a general class of discontinuous modulation function. Another additional application of renormalization techniques occurs in the study of strange non-chaotic attractors (SNAs). We investigate the non-smooth pitchfork bifurcation route to SNA in systems of \pinched skew-product" type and give conditions for self-similar behaviour of the attractor at the critical point of transition. In addition, we describe how the attractor scales as we approach a bifurcation curve. To conclude, we study the box-counting dimension of strange non-chaotic attractors (SNAs) created by this bifurcation. We provide compelling evidence that a non-critical SNA has dimension 2. The method we adopt becomes more accurate in the study of piecewise linear SNAs. We also provide numerical evidence that the dimension of a critical SNA is not necessarily equal to 2, but can lie between 1 and 2.

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.

This thesis details research I have carried out in the field of quantum walks, which are the quantum analogue of classical random walks. Quantum walks have been shown to offer a significant speed-up compared to classical random walks for certain tasks and for this reason there has been considerable interest in their use in algorithmic settings, as well as in experimental demonstrations of such phenomena. One of the most interesting developments in quantum walk research is their application to spatial searches, where one searches for a particular site of some network or lattice structure. There has been much work done on the creation of discrete- and continuous-time quantum walk search algorithms on various lattice types. However, it has remained an issue that continuous-time searches on two-dimensional lattices have required the inclusion of additional memory in order to be effective, memory which takes the form of extra internal degrees of freedom for the walker. In this work, we describe how the need for extra degrees of freedom can be negated by utilising a graphene lattice, demonstrating that a continuous-time quantum search in the experimentally relevant regime of two-dimensions is possible. This is achieved through alternative methods of marking a particular site to previous searches, creating a quantum search protocol at the Dirac point in graphene. We demonstrate that this search mechanism can also be adapted to allow state transfer across the lattice. These two processes offer new methods for channelling information across lattices between specific sites and supports the possibility of graphene devices which operate at a single-atom level. Recent experiments on microwave analogues of graphene that adapt these ideas, which we will detail, demonstrate the feasibility of realising the quantum search and transfer mechanisms on graphene.

The Totally Asymmetric Simple Exclusion Process (TASEP) is often considered one of the fundamental models of non-equilibrium statistical mechanics, due to its well understood steady state and the fact that it can exhibit condensation, phase separation and phase transitions in one spatial dimension. As a minimal model of traffic flow it has enjoyed many applications, including the transcription of proteins by ribosomal motors moving along an mRNA track, the transport of cargo between cells and more human-scale traffic flow problems such as the dynamics of bus routes. It consists of a one-dimensional lattice of sites filled with a number of particles constrained to move in a particular direction, which move to adjacent sites probabilistically and interact by mutual exclusion. The study of non-Markovian interacting particle systems is in its infancy, due in part to a lack of a framework for addressing them analytically. In this thesis we extend the TASEP to allow the rate of transition between sites to depend on how long the particle in question has been stationary by using non-Poissonian waiting time distributions. We discover that if the waiting time distribution has infinite variance, a dynamic condensation effect occurs whereby every particle on the system comes to rest in a single traffic jam. As the lattice size increases, so do the characteristic condensate lifetimes and the probability that a condensate will interact with the preceding one by forming out of its remnants. This implies that the thermodynamic limit depends on the dynamics of such spatially complete condensates. As the characteristic condensate lifetimes increase, the standard continuous time Monte Carlo simulation method results in an increasingly large fraction of failed moves. This is computationally costly and led to a limit on the sizes of lattice we could simulate. We integrate out the failed moves to create a rejection-free algorithm which allows us to see the interacting condensates more clearly. We find that if condensates do not fully dissolve, the condensate lifetime ages and saturates to a particular value. An unforeseen consequence of this new technique, is that it also allowed us to gain a mathematical understanding of the ageing of condensates, and its dependence on system size. Using this we can see that the fraction of time spent in the spatially complete condensate tends to one in the thermodynamic limit. A random walker in a random force field has to escape potential wells of random depth, which gives rise to a power law waiting time distribution. We use the non-Markovian TASEP to investigate this model with a number of interacting particles. We find that if the potential well is re-sampled after every failed move, then this system is equivalent to the non-Markovian TASEP. If the potential well is only re-sampled after a successful move, then we restore particle-hole symmetry, allow condensates to completely dissolve, and the thermodynamic limit spends a finite fraction of time in the spatially complete state. We then generalised the non-Markovian TASEP to allow for particles to move in both directions. We find that the full condensation effect remains robust except for the case of perfect symmetry.

This thesis investigates the dynamical properties of equilibrium and nonequilibrium systems, both quantum and classical, under the guise of a thermodynamic formalism. Large deviation functions associated with the generating functions of time-integrated observables play the role of dynamical free energies and thus determine the trajectory phase structure of a system. The 1d Glauber-Ising chain is studied using the time-integrated energy as the dynamical order parameter and a whole curve of second order trajectory transitions are uncovered in the complex counting field plane. The leading dynamical Lee-Yang zeros of the associated generating function are extracted directly from the time dependent high order cumulants. Resolving the cumulants into constituent contributions the motion of each contribution’s leading Lee-Yang zeros pair allows one to infer the positions of the trajectory transition points. Contrastingly if one uses the full cumulants only the positions of those closest to the origin, in the limit of low temperatures, can be inferred. Motivated by homodyne detection schemes this thermodynamic approach to trajectories is extended to the quadrature trajectories of light emitted from open quantum systems. Using this dynamical observable the trajectory phases of a simple “blinking” 3-level system, two weakly coupled 2-level systems and the micromaser are studied. The trajectory phases of this observable are found to either carry as much information as the photon emission trajectories or in some cases capture extra dynamically features of the system (the second example). Finally, the statistics of the time-integrated longitudinal and transverse magnetization in the 1d transverse field quantum Ising model are explored. In both cases no large deviation function exists but the generating functions are still calculable. From the singularities of these generating functions new transition lines emerge. These were shown to be linked to: (a) the survival probability of an associated open system, (b) PT-symmetry, (c) the temporal scaling of the cumulants and (d) the topology of an associated set of states.

In this Thesis I present simulation- and theory-based studies of pattern formation and growth in collections of micro-organisms, in particular bacterial colonies. The aim of these studies is to introduce simple models of the 'micro-scale' behaviour of bacterial cells in order to study the emergent behaviour of large collections of them. To do this, computer simulations and theoretical techniques from statistical physics, and in particular non-equilibrium statistical physics, were used, as the systems under study are far from thermodynamic equilibrium, in common with most biological systems. Since the elements making up these sytems - the micro-organisms - are active, constantly transducing energy from their environment in order to move and grow, they can be viewed as `active matter' systems. First, I describe my work on a generalization of an archetypal model of active matter - the Vicsek model of flocking behaviour - in which the speed of motion of active particles depends on the local density of particles. Such an interaction had previously been shown to be responsible for some forms of pattern formation in bacterial colonies grown on agar plates in the laboratory. Simulations and theory demonstrated a variety of pattern formation in this system, and these results may be relevant to explaining behaviour observed in experiments done on collections of molecular motors and actin fibres. I then go on to describe work on modelling pattern formation and growth in bacterial biofilms - dense colonies of cells growing on top of solid surfaces. I introduce a simple simulation model for the growth of non-motile cells on a flat surface, whereby they move only by growing and pushing on each other as they grow. Such colonies have previously been observed experimentally to demonstrate a transition from round to 'branched' colonies, with a pattern similar to diffusion-limited aggregation. From these simulations and analytical modelling, a theory of the growth of such colonies is developed which is quite different from previous theories. For example, I find that the colony cannot grow at a constant speed if the cells are not compressible. Finally, I present some results on genetic drift and evolution in growing bacterial colonies. Genetic drift is greatly enhanced in colonies which are expanding in space, as only a few individuals at the edge of the population are able to pass on their genes onto their progeny. The individual-based simulations of biofilms described above are used to analyse which factors - such as the shape of the colony, the thickness of the growing layer of cells, and the interactions between the cells - affect the rate of genetic drift and the probability of fixation of beneficial mutations. This has implications, for example, for the evolution of antibiotic resistance in such colonies.

This thesis utilises large deviation methods to study nonequilibrium phenomena in both quantum and classical systems. The dynamical analogues of the ensembles of statistical mechanics are used to explore dynamical phase spaces of systems, quantifying atypical fluctuations that can play a critical role in long term behaviour. A dynamical ensemble based on fixed numbers of dynamical events, allowing trajectory observation time to fluctuate, is introduced. This ensemble, denoted the x-ensemble, is found to be well suited to numerically simulate atypical fluctuations using transition path sampling (TPS). x-ensemble TPS schemes are analysed with reference to existing methods in both quantum and classical stochastic systems, and are found to offer more flexibility and efficiency in a variety of situations. The potential to develop this scheme into a self-optimizing algorithm is discussed with examples. The x-ensemble is then used in three non-equilibrium scenarios. Firstly in plaquette models of glass formers, in an effort to provide insight into the nature of the glass transition. It is shown that a two-dimensional triangular plaquette model (TPM) exhibits both a trajectory phase-transition between dynamical active and inactive phases, and when two replicas are coupled, a thermal phase transition between states of low and high overlap between the replicas. These two transitions are similar to those seen to occur in more realistic glass formers. When the TPM is generalised to a three-dimensional square pyramid plaquette model (SPyM) these dynamical and thermodynamic features of interest remain. It is argued that these models therefore provide an ideal test-bed for competing theories of the glass transition. Secondly the x-ensemble is used to define and analyse the dynamical analogue of the Jarzynski equality, allowing for the computation of dynamical free energy differences with, in principle, arbitrarily fast protocols linking two dynamical states. This relation is tested and found to hold in open quantum systems. Finally the partition sum zeros method of Lee and Yang is used to extract the location of dynamical phase transitions from the high-order, short-time cumulants of the x-ensemble. Results in both classical and open quantum systems are compared with previously studied dynamical ensembles, providing insight into the nature in which dynamical behaviours are encoded by these ensembles.

Orientador: Adonai S. Sant'Anna / Dissertação (mestrado) - Universidade Federal do Paraná / Resumo: Este trabalho é uma tese em fundamentos da física. Aqui se dá atenção a certos aspectos da mecânica quântica, usando-se uma matemática não-clássica. Usa-se aqui teoria de quase-conjuntos.A teoria dos quase-conjuntos (ou simplesmente q-conjuntos) generaliza a teoria usual de conjuntos de Zermelo-Fraenkel, permitindo a existência de conjuntos de elementos sem individualidade. Como contribuição deste trabalho, apresenta-se aqui uma formulação para q-conjuntos que permite uma combinatória q-conjuntista suficientemente rica para tratarmos de certas questões relativas às estatísticas usuais em física.Deduzem-se, neste contexto, as estatísticas usualmente empregadas em mecânica estatística. Como resultado, tanto as estatísticas quânticas quanto a de Maxwell-Boltzmânn são obtidas sem a necessidade de se admitir que as partículas de um dado gás tenham individualidade. / Abstract: This is a Masters thesis dissertation on the foundations of Physics. Some statistical aspects of Quantum Mechanics are treated by using non-classical mathematics,namely, Quasi-set Theory.Quasi-set theory (q-set theory, for short) generalizes the usual Zermelo-Fraenkel set theory by enconpassing sets of entities without identity. As a contribuition of this work, q-set theory is given here an approach whose combinatorics can deal with certain aspects concerning the usual statistics in physics.Some of the statistics are then derived in this framework. As a result, the quantum statistics, as well as Maxwell-Boltzmann statistics, are retrieved without the assumption that the elements which make up the system under consideration (say, a given gas) are endowed with identity.