Dissertations / Theses on the topic 'Lattice gas system'

We explore steady-state properties of a driven lattice gas, which is a simple model of interacting many-particle systems, driven far from equilibrium by an external field. First, we study a system on a square lattice with periodic boundary conditions (PBC) along both principal lattice axes, while the drive acts along only one of these axes. For such systems, we analyze the full distribution of structure factors. Next, we investigate the effects of imposing other boundary conditions on the system. In particular, we focus on models with shifted periodic boundary conditions (SPBC) along one axis and open boundary conditions (OBC) along the other axis. The OBC allow us to have a steady flux of particles through the system while the SPBC permits us to drive the system in a range of possibilities. Using Monte Carlo simulation techniques, we discover a rich variety of phenomena, especially at low temperatures. A continuum theory for the densities, based on Langevin equations, is formulated and its predictions compared to simulation data. Many large scale properties are described successfully.
Ph. D.

In recent years a new interesting research avenue has emerged in non-equilibrium statistical physics, namely studies of collective responses in spatially inhomogeneous systems. Whereas substantial progress has been made in understanding the origins and the often universal nature of cooperative behavior in systems far from equilibrium, it is still unclear whether it is possible to control their global collective stochastic dynamics through local manipulations. Therefore, a comprehensive characterization of spatially inhomogeneous non-equilibrium systems is required. In the first system, we explore a variant of the Katz–Lebowitz–Spohn (KLS) driven lattice gas in two dimensions, where the lattice is split into two regions that are coupled to heat baths with distinct temperatures T > Tc and Tc respectively, where Tc indicates the critical temperature for phase ordering. The geometry was arranged such that the temperature boundaries are oriented perpendicular or parallel to the external particle drive and resulting net current. For perpendicular orientation of the temperature boundaries, in the hotter region, the system behaves like the (totally) asymmetric exclusion processes (TASEP), and experiences particle blockage in front of the interface to the critical region. This blockage is induced by extended particle clusters, growing logarithmically with system size, in the critical region. We observe the density profiles in both high- and low-temperature subsystems to be similar to the well-characterized coexistence and maximal-current phases in (T)ASEP models with open boundary conditions, which are respectively governed by hyperbolic and trigonometric tangent functions. Yet if the lower temperature is set to Tc, we detect marked fluctuation corrections to the mean-field density profiles, e.g., the corresponding critical KLS power-law density decay near the interfaces into the cooler region. For parallel orientation of the temperature boundaries, we have explored the changes in the dynamical behavior of the hybrid KLS model that are induced by our choice of the hopping rates across the temperature boundaries. If these hopping rates at the interfaces satisfy particle-hole symmetry, the current difference across them generates a vector flow diagram akin to an infinite flat vortex sheet. We have studied the finite-size scaling of the particle density fluctuations in both temperature regions, and observed that it is controlled by the respective temperature values. If the colder subsystem is maintained at the KLS critical temperature, while the hotter subsystem's temperature is set much higher, the interface current greatly suppresses particle exchange between the two regions. As a result of the ensuing effective subsystem decoupling, strong fluctuations persist in the critical region, whence the particle density fluctuations scale with the KLS critical exponents. However, if both temperatures are set well above the critical temperature, the particle density fluctuations scale according to the totally asymmetric exclusion process. We have also measured the entropy production rate in both subsystems; it displays intriguing algebraic decay in the critical region, while it saturates quickly at a small but non-zero level in the hotter region. The second system is a lattice gas that simulates the spread of COVID-19 epidemics using the paradigmatic stochastic Susceptible-Infectious-Recovered (SIR) model. In our effort to control the spread of the infection of a lattice, we robustly find that the intensity and spatial spread on the epidemic recurrence wave can be limited to a manageable extent provided release of these restrictions is delayed sufficiently (for a duration of at least thrice the time until the peak of the unmitigated outbreak).
Doctor of Philosophy
In recent years a new interesting research avenue has emerged in far-from-equilibrium statistical physics, namely studies of collective behavior in spatially non-uniform systems. Whereas substantial progress has been made in understanding the origins and the often universal nature of cooperative behavior in systems far from equilibrium, it is still unclear whether it is possible to control their global collective and randomly determined dynamics through local manipulations. Therefore, a comprehensive characterization of spatially non-uniform systems out of equilibrium is required. In the first system, we explore a variant of the two-dimensional lattice gas with completely biased diffusion in one direction and attractive particle interactions. By lattice gas we mean a lattice filled with particles that can hop on nearest-neighbor empty sites. The system we are considering is a lattice that is split into two regions, which in turn are maintained at distinct temperatures T > Tc and Tc, respectively, with Tc indicating the critical temperature for the second-order phase transition. The geometry of the lattice was arranged such that the temperature boundaries are oriented perpendicular or parallel to the external particle drive that is responsible for a completely biased diffusion. When the temperature boundaries are oriented perpendicular to the drive, in the hotter region with temperature T > Tc, the system evolves as if there are no attractive interactions between the particles, and experiences particle blockage in front of the temperature boundary from the hotter region held at T>Tc to the critical region held at Tc. This accumulation of particles at the temperature boundary is induced by elongated collections of particle, i.e., particle clusters in the critical region. We observe the particle density profiles (density(x) vs x plots) in both high-and low-temperature subsystems to be similar to the density profiles found for other well-characterized (T)ASEP models with open boundary conditions, which are in the coexistence and maximal-current phases, and which are respectively governed by hyperbolic and trigonometric tangent functions. Yet if the lower temperature is set to Tc, we detect marked corrections to the hyperbolic and trigonometric tangent-like density profiles due to fluctuations, e.g., we observe the algebraic power-law decay of the density near the interfaces into the cooler region with the critical KLS exponent. For a parallel orientation of the temperature boundaries, we have explored the changes in the particle dynamics of the two-temperature KLS model that are induced by our choice of the particle hopping rates across the temperature boundaries. If these particle hopping rates at the temperature interfaces satisfy particle-hole symmetry (i.e. remain unchanged when particles are replaced with holes and vice versa), the particle current difference across them generates a current vector flow diagram akin to an infinite flat vortex sheet. We have studied how the particle density fluctuations in both temperature regions scale with the system size, and observed that the scaling is controlled by the respective temperature values. If the colder subsystem is maintained at the KLS critical temperature Tcold = Tc, while the hotter subsystem's temperature is set much higher Thot >> Tc, the particle currents at the interface greatly suppresses particle exchange between the two temperature regions. As a result of the ensuing effective subsystem separation from each other, strong fluctuations persist in the critical region, whence the particle density fluctuations scale with the KLS critical exponents. However, if both temperatures are set well above the critical temperature, the particle density fluctuations scale with different scaling exponents, that fall into the totally asymmetric exclusion process (TASEP) universality class. We have also measured the rate of the entropy production in both subsystems; it displays intriguing algebraic decay in the critical region, while it reaches quickly a small but non-zero value in the hotter region. The second system is a lattice filled with particles of different types that hop around the lattice and are subjected to different sorts of reactions. That process simulates the spread of the COVID-19 epidemic using the paradigmatic random-process-based Susceptible-Infectious-Recovered (SIR) model. In our effort to control the spread of the infection of a lattice, we robustly find that the intensity and spatial spread of the epidemic second wave can be limited to a manageable extent provided release of these restrictions is delayed sufficiently (for a duration of at least thrice the time until the peak of the unmitigated outbreak).

Nucleation is the first step in the formation of a new phase in a thermodynamic system. The Classical Nucleation Theory (CNT) is the traditional theory used to describe this phenomenon. The object of this thesis is to investigate nucleation beyond one of the most significant limitations of the CNT: the assumption that the surface tension of a nucleating cluster of the new phase is independent of the cluster’s size and has the same value that it would have in the bulk of the new phase. In order to accomplish this, we consider a microscopic, two-dimensional Lattice Gas Automata (LGA) model of precipitate nucleation in a supersaturated system, with model input parameters Ess (solid particle-to-solid particle bonding energy), Esw (solid particle-to-water particle bonding energy), η (next-to-nearest neighbour bonding coeffiicent in solid phase), and Cin (initial solute concentration). The LGA method was chosen for its advantages of easy implementation, low memory requirements, and fast computation speed. Analytical results for the system’s concentration and the crystal radius as functions of time are derived and the former is fit to the simulation data in order to determine the system’s equilibrium concentration. A mean first-passage time (MFPT) technique is used to obtain the nucleation rate and critical nucleus size from the simulation data. The nucleation rate and supersaturation are evaluated using a modification to the CNT that incorporates a two-dimensional, radius-dependent surface tension term. The Tolman parameter, δ, which controls the radius-dependence of the surface tension, decreases (increases) as a function of the magnitude of Ess (Esw), at fixed values of η and Esw (Ess). On the other hand, δ increases as η increases while Ess and Esw are held constant. The constant surface tension term of the CNT, Σ0, increases (decreases) with increasing magnitudes of Ess (Esw) fixed values of Esw (Ess), and increases as η is increased. Together, these results indicate an increase in the radius-dependent surface tension, Σ, with respect to increasing magnitude of Ess relative to the magnitude of Esw. Σ0 increases linearly as a function of the change in energy during an attachment or detachment reaction, |ΔE|, however with a slope less than that predicted for a crystal that is uniformly packed at maximum density.

The main topic of this thesis is the cluster expansion technique and its applica- tions to a variety of problems ranging from probability to physics and chemistry. The thesis is divided into a first part of relevant known results from the literature, and a second part where we present our contribution. We start by recalling some central aspects of the cluster expansion, and hence, general cluster expansion theorems in the grand-canonical and canonical ensem- bles and related results. Then, we present a classical problem in probability about computing large and moderate deviations as well as its formulation in statistical mechanics in the canonical/micro-canonical and the canonical/grand-canonical ensembles. We consider both the case of continuous - in R^d - and discrete - in Z^d - systems of interacting particles. In the second part, we present our results. First, we consider a system of classical particles confined in a box Λ ⊂ Rd with zero boundary conditions in- teracting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical partition function in the high temperature - low density regime, we prove moderate and precise large deviations from the mean value of the number of particles with respect to the grand-canonical Gibbs measure. In this way we have a direct method of computing both the exponential rate as well as the pre-factor and obtain explicit error terms. Estimates compar- ing with the infinite volume versions of the above are also provided. Second, we show the validity of the cluster expansion in the canonical ensemble for the Ising model. We compare the lower bound of its radius of convergence with the one computed by the virial expansion working in the grand-canonical ensemble. Us- ing the cluster expansion we give direct proofs with quantification of the higher order error terms for the decay of correlations, and also in this case, for central limit theorem and large deviations. In the last part of the thesis, using a strategy given in the literature in the grand-canonical ensemble, we perform the cluster expansion for colloids in the canonical ensemble, considering periodic boundary conditions. The novelty consists in the fact that we establish a hierarchy in the order of integration, which allows to work with the effective system.

We explore the nature of driven stochastic lattice systems with non-periodic boundary conditions. The systems consist of particle and holes which move by exchanges of nearest neighbor particle-hole pairs. These exchanges are controlled by the energetics associated with an internal Hamiltonian, an external drive and a stochastic coupling to a heat reservoir. The effect of the drive is to bias particle-hole exchanges along the field in such a way that a particle current can be established. Hard-core volume constraints limit the occupation of only one particle (hole) per lattice site. For certain regimes of the overall particle density and temperature, a system displays a homogeneous disordered phase. We investigate cooperative behavior in this phase by using two-point spatial correlation functions and structure factors. By varying the particle density and the temperature, the system orders into a phase separated state, consisting of particle-rich and particle-poor regions. The temperature and density for the co-existence state depend on the boundary conditions. By using Monte Carlo simulations, we establish co-existence curves for systems with shifted periodic boundary conditions.
Ph. D.

We study the non-equilibrium behavior of systems that are driven out of equilibrium from the interface. In the first part of this thesis, we study a model of a two-dimensional lattice gas that is in contact with two heat baths that are at different temperatures. Performing Monte Carlo simulations, we find that there are three possible types of non-equilibrium steady states, depending on the values of certain system parameters. They include a disordered phase, a fully phase separated state, and an interesting state with striped patterns in the half of the lattice where the temperature is lower. The last one is a novel non-equilibrium steady state that we study systematically by varying the system parameters. To obtain the non-equilibrium finite-size phase diagram, we perform a spectrum analysis to classify not only the three major states, but also the sub-states of the striped phase. In the second part of the thesis, we study magnetic friction that results when two Potts systems move with respect to each other. In this research, we first study a model that consists of two interacting Potts blocks, where one block moves on top of the other. As a result, the system is driven out of equilibrium constantly. In our research we find for weak interfacial couplings that the contacting surfaces behave rather similar to a free surface. If the interfacial coupling is strong, however, anisotropic spin patterns appear on the contacting surfaces. This study is extended to a three-dimensional Potts wedge with a tip sliding along the surface of a Potts block. It is found that the shape of the Potts lattice influences the surface behavior of the system.
Ph. D.

In this work we provide several improvements in the study of phase transitions of interacting particle systems: - We determine a quantitative relation between non-extremality of the limiting Gibbs measure of a tree-based spin system, and the temporal mixing of the Glauber Dynamics over its finite projections. We define the concept of 'sensitivity' of a reconstruction scheme to establish such a relation. In particular, we focus on the independent sets model, determining a phase transition for the mixing time of the Glauber dynamics at the same location of the extremality threshold of the simple invariant Gibbs version of the model. - We develop the technical analysis of the so-called spatial mixing conditions for interacting particle systems to account for the connectivity structure of the underlying graph. This analysis leads to improvements regarding the location of the uniqueness/non-uniqueness phase transition for the independent sets model over amenable graphs; among them, the elusive hard-square model in lattice statistics, which has received attention since Baxter's solution of the analogous hard-hexagon in 1980. - We build on the work of Montanari and Gerschenfeld to determine the existence of correlations for the coloring model in sparse random graphs. In particular, we prove that correlations exist above the 'clustering' threshold of such a model; thus providing further evidence for the conjectural algorithmic 'hardness' occurring at such a point.

Dans cette thèse, nous étudions la robustesse dans le contexte de la modélisation de systèmes complexes par les automates cellulaires. En effet, si l'on cherche à reproduire un comportement émergent à partir d'un modèle d'automate cellulaire, il nous semble nécessaire de se demander si les comportements observés sont bien le résultat d'interactions entre entités constituantes, ou bien s'ils dépendent d'une définition particulière du modèle. Nous allons ainsi être amenés à considérer la robustesse du modèle, à savoir la résistance de son comportement à de petites variations sur les attributs de sa définition. Dans un premier temps, nous montrons la pertinence de cette approche en considérant plusieurs définitions possibles d'une perturbation de la mise à jour globale et en les appliquant à une classe simple et représentative de modèles d'automates cellulaires, les Automates Cellulaires Elémentaires. Nous observons que, malgré le fait que nos perturbations soient proches et qu'une majorité des modèles considérés ne change pas de comportement, quelques cas particuliers montrent des changements qualitatifs du comportement que nous étudions plus en détail. Dans un second temps, nous appliquons cette approche en nous penchant sur un modèle particulier d'automate cellulaire, qui simule le phénomène de formation d'essaim à partir d'un modèle évolué d'automate cellulaire, le gaz sur réseau. Nous explorons la robustesse du comportement du modèle en considérant la perturbation de deux attributs du modèle, la forme de la grille cellulaire et la mise à jour globale, et en tirons les conclusions sur la relation entre l'observation du comportement et la définition précise du modèle.

Two dimensional (2D) systems with low carrier density is an outstanding platform for studying a wide spectrum of physics. These include both classical and quantum effects, arising from disorder, Coulomb interactions and even non-trivial topological properties of band-structure. In this thesis, we have explored the physics at low carrier number density in GaAs/AlGaAs heterostructure and bilayer graphene, by investigating in a larger phase space using a variety of electrical measurement tools. A two-dimensional electron system (2DES) formed in a GaAs/AlGaAs heterostructure offers an avenue to build a variety of mesoscopic devices, primarily because its surface gates can very effectively control its carrier density profile. In the first half of the thesis, we study the relevance of disorder in two kinds of devices made in a 2DES. A very strong negative gate voltage not only reduces the carrier density of the 2DES, but also drives it to a disordered state. In this state, we explore a new direction in parameter space by increasing in-plane electric field and investigating its magneto-resistance (MR). At sufficiently strong gate voltage and source-drain bias, we discover a remarkably linear MR. Its enormous magnitude and weak temperature dependence indicate that this is a classical effect of disorder. In another study, we examine a specially designed dual-gated device that can induce low number density in a periodic pattern. By applying appropriate gate voltages, we demonstrate the formation of an electrostatically tunable quantum dot lattice and study the impact of disorder on it. This work is important in paving way for solid state based platform for experimental simulations of artificial solids. The most striking property of bilayer graphene is the ability to open its band gap by a perpendicular electric field, giving the prospects of enabling a large set of de-vice applications. However, despite a band gap, a number of transport mechanisms are still active at very low densities that range from hopping transport through bulk to topologically protected 1D transport at the edges or along 1D crystal dislocations. In the second half of the thesis, we have used higher order statistical moment of resistance/conductance fluctuations, namely the variance of the fluctuations, to complement averaged resistance/conductance, and study and infer the dominant transport mechanism at low densities in a gapped bilayer graphene. Our results show possible evidence of percolative transport and topologically protected edge transport at different ranges of low number densities. We also explore the same phase space by studying its mesoscopic conductance fluctuations at very low temperatures. This is the first of its kind systematic experiment in a dual-gated bilayer graphene device. Its conductance fluctuations have several anomalous features suggesting non-universal behaviour which is at odds with conventional disordered systems.

The work presented in this thesis concerns the study of quantum many-body physics by making use Bose-Einstein condensates loaded in optical lattices potentials. The first part describes the development of a new experimental strategy for the production of the degenerate atomic sample, the second part concerns the optimal control ground state production and the entanglement characterization on a systems of interacting Bosons across the superfluid - Mott insulator quantum phase transition, and the third part illustrates the study of the dynamical properties of an array of 1D gases performed via Bragg spectroscopy.