While the statistical mechanics of systems in thermal equilibrium is a well established discipline, nonequilibrium systems are fundamentally much less well understood, even though most natural phenomena fall into the latter category. In particular, there is as yet no nonequilibrium analog for the systematic formalism of Gibbs ensembles. Rather than deal with the difficult problem of general nonequilibrium systems, this study is restricted to the steady states of a simple model whose equilibrium properties are well known. The Ising lattice gas displays a number of surprising phenomena when driven into nonequilibrium steady states. This study extends previous work to a more general model with anisotropic interparticle interactions. Using Monte Carlo simulations, we obtain the phase diagram for the model, controlled by the driving field, temperature, and anisotropy parameter α. Under saturation drive, the shift in the transition temperature between ordered and disordered states can be either positive or negative, depending on α ≡ √(𝐽_∥/𝐽_⟂). The possible existence at large α of an additional phase ordered in only one direction is discussed. For finite drives, both first and second order transitions are observed. A novel technique for locating the first order transition line is presented. Some aspects of the phase diagram can be predicted by investigating the two-point correlation function to first order in a high temperature series expansion. However, the series expansion fails to predict even qualitatively the α-dependence of the critical temperature. / Master of Science

We study the phase diagram of an interacting lattice gas of two species of particles and holes, driven out of equilibrium by a local hopping bias (denoted by `E'). Particles interact by excluded volume and nearest-neighbor attractions. We present a detailed Monte Carlo investigation of the phase diagram. Three phases are found, with a homogenous phase at high temperatures and two distinct ordered phases at lower temperatures. Which ordered phase is observed depends on the parameter f, which controls the ratio of the two types of particles. At small f, there is nearly a single species, and a transition is observed into a KLS-type ordered phase. At larger f, the minority species are sufficiently dense to form a transverse blockage, and a sequence of two transitions are observed as the temperature is lowered. First, a continuous boundary is crossed into an SHZ-type ordered phase, then at a lower temperature a first-order boundary is crossed into the KLS-type ordered phase. At some critical value of f is a bicritical point, where the first-order line branches from the two continuous boundaries. We also consider correlations in the homogenous phase, by constructing a continuum description and comparing to the results of simulations. Long range correlations are present in both the theoretical results and the simulations, though certain details of the theory do not fit the observations very well. Finally, we examine the beahvior of three-point correlations in the single-species (KLS) limit. Nontrivial three-point correlations are directly related to the nonzero bias E. We therefore consider the behavior of the three-point correlations as a function of E. We find that the three-point signal saturates very rapidly with E. There are some difficulties interpreting the data at small E. / Ph. D.

A lattice gas with equal numbers of oppositely charged particles, diffusing under the influence of a uniform electric field and an excluded volume condition undergoes an order-disorder phase transition, controlled by the particle density and the field strength. This transition may be continuous (second order) or continuous (first order). Results from previous discrete simulations are shown, and a theoretical continuum model is developed. As this is a nonequilibrium system, there is no associated free energy to determine the location of a first order transition. Instead, the model equations for this system are evolved in time numerically, and the locus of this transition is determined via the presence of a stable state with coexisting regions of order and disorder. The Crank-Nicholson, nonlinear Gauss-Seidel, and GMRES algorithms used to solve the model equations are discussed. Performance enhancements and limits on convergence are considered. / Master of Science

We numerically investigate the long-time behavior of the density-density auto-correlation function in driven lattice gases, with particle exclusion and periodic boundary conditions in one, two, and three dimensions using precise Monte Carlo simulations of larger system sizes than previous studies. In the one-dimensional asymmetric exclusion process on a ring with half the lattice sites occupied, we find that correlations induce extremely slow relaxation to the asymptotic power law decay We compare the crossover functions obtained from our simulations with various analytic results in the literature, and analyze the characteristic oscillations that occur in finite systems away from half-filling. As expected, correlations are weak in three dimensions and consequently the mean-field description is adequate. We also investigate the relaxation towards the non-equilibrium steady state in the two-time density-density auto-correlations, starting from strongly correlated initial conditions. We obtain simple aging scaling behavior in one, two, and three dimensions, with the expected power laws. We numerically investigate the behavior of driven lattice gases with nearest neighbor interactions at half-filling with periodic boundary conditions below and at the critical temperature using Monte Carlo simulations of very large lattices in two dimensions. This work is one of few that explores the relaxation to a non-equilibrium steady state. We obtain data collapse for the finite-size scaling form of density-density auto-correlation function at the critical point. We achieve data collapse using finite-size scaling of the time-dependent order parameter during the transient regime starting from strongly correlated initial conditions. We present simple aging scaling of the density-density auto-correlation function at the critical point starting from strongly correlated initial conditions using Monte Carlo simulations of two different lattice anisotropies. We thus unambiguously confirm the critical exponents determined by renormalization group methods using measurement of dynamic quantities in the transient regime. Measuring these dynamic quantities in the transient regime provides more conclusive measurements of the critical exponents than previous studies measuring static quantities in the stationary state. We provide qualitative arguments that the lattice anisotropy determines the steady-state for sub-critical quenches. / Ph. D.

We examine the effect of lane preference on a quasi one-dimensional three-state driven lattice gas, consisting of holes and positive and negative particles, and periodic boundary conditions in the longitudinal direction. Particles move via particle-hole and, with a lesser rate, particle-particle exchanges; the species are driven in opposite directions along the lattice, each preferring one of the lanes with a given probability, <I>p</I>. The model can be interpreted as traffic flow on a two-lane beltway, with fast cars preferring the left lane and slow cars preferring the right, viewed in a comoving frame. In steady-sate, the system typically exhibits a macroscopic cluster containing a majority of the particles. At very high values of <I>p</I>, a first order transition takes the system to a spatially disordered state. Using Monte Carlo simulations to analyze the system, we find that the size of the cluster increases with lane preference. We also observe a region of negative response, where increasing the lane preference <I>decreases</I> the number of particles in their favored lane, against all expectations. In addition, simulations show an intriguing sequence of density profiles for the two species. We apply mean-field theory, continuity equations, and symmetries to derive relationships between observables to make a number of predictions verified by the Monte Carlo data. / Master of Science

We investigate the dynamics of a three-state stochastic lattice gas, consisting of holes and two oppositely "charged" species of particles, under the influence of an "electric" field, at zero total charge. Interacting only through an excluded volume constraint, particles can hop to nearest neighbor empty sites, but particle-particle exchange between oppositely charged particles is also allowed on a separate time scale. Controlled by this relative time scale, particle density and drive, the system orders into a charge-segregated state. Using a combination of Monte Carlo simulations and continuum field theory techniques, we study the order of these transitions and map out the steady state phase diagram of the system. On a single sheet of transitions, a line of multicritical points is found, separating the first order and continuous transitions. Furthermore, we study the steady-state structure factors in the disordered phase where homogeneous configurations are stable against small harmonic perturbations. The average structure factors show a discontinuity singularity at the origin which in real space predicts an intricate crossover between power laws of different kinds. We also seek for generic statistical properties of these quantities. The probability distributions of the structure factors are universal asymmetric exponential distributions. This research was supported in part by grants from the National Science Foundation through the Division of Materials Research. / Ph. D.

Phase segregated states in a simple three-state stochastic lattice gas are investigated. A two dimensional finite lattice with periodic boundary conditions is filled with one hole and two oppositely "charged" species of particles, subject to an excluded volume constraint. Starting from a completely disordered initial configuration, a sufficiently large external "electric" field <I>E</I> induces the phase segregation, by separating the charges into two strips and "trapping" the hole at an interface between them. Focusing on the steady state, the scaling properties of an appropriate order parameter, depending on drive and system size, are investigated by mean-field theory and Monte Carlo methods. Density profiles of the two interfaces in the ordered system are studied with the help of Monte Carlo simulations and are found to scale in the field-dependent variable, Ε = 2 tanh <I>E</I> /2), for <I>E</I> ≲ 0.8. For larger values of <I>E</I>, independent approximations of the interfacial profiles, obtained within the framework of mean-field theory, exhibit significant deviations from the Monte Carlo data. Interestingly, the deviations can be reduced significantly by a slight modification of the mean-field theory. / Master of Science