VTechWorks Repository :: Browsing by Author "Zia, Royce K. P."

Browsing by Author "Zia, Royce K. P."

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Analytic Results for Hopping Models with Excluded Volume Constraint
Toroczkai, Zoltan (Virginia Tech, 1997-09-04)
Part I: The Theory of Brownian Vacancy Driven Walk We analyze the lattice walk performed by a tagged member of an infinite 'sea' of particles filling a d-dimensional lattice, in the presence of a single vacancy. The vacancy is allowed to be occupied with probability 1/2d by any of its 2d nearest neighbors, so that it executes a Brownian walk. Particle-particle exchange is forbidden; the only interaction between them being hard core exclusion. Thus, the tagged particle, differing from the others only by its tag, moves only when it exchanges places with the hole. In this sense, it is a random walk "driven" by the Brownian vacancy. The probability distributions for its displacement and for the number of steps taken, after n-steps of the vacancy, are derived. Neither is a Gaussian! We also show that the only nontrivial dimension where the walk is recurrent is d=2. As an application, we compute the expected energy shift caused by a Brownian vacancy in a model for an extreme anisotropic binary alloy. In the last chapter we present a Monte-Carlo study and a mean-field analysis for interface erosion caused by mobile vacancies. Part II: One-Dimensional Periodic Hopping Models with Broken Translational Invariance.Case of a Mobile Directional Impurity We study a random walk on a one-dimensional periodic lattice with arbitrary hopping rates. Further, the lattice contains a single mobile, directional impurity (defect bond), across which the rate is fixed at another arbitrary value. Due to the defect, translational invariance is broken, even if all other rates are identical. The structure of Master equations lead naturally to the introduction of a new entity, associated with the walker-impurity pair which we call the quasi-walker. Analytic solution for the distributions in the steady state limit is obtained. The velocities and diffusion constants for both the random walker and impurity are given, being simply related to that of the quasi-particle through physically meaningful equations. As an application, we extend the Duke-Rubinstein reputation model of gel electrophoresis to include polymers with impurities and give the exact distribution of the steady state.
Anomalous nucleation far from equilibrium
Georgiev, I. T.; Schmittmann, Beate; Zia, Royce K. P. (American Physical Society, 2005-03-25)
We present precision Monte Carlo data and analytic arguments for an asymmetric exclusion process, involving two species of particles driven in opposite directions on a 2xL lattice. To resolve a stark discrepancy between earlier simulation data and an analytic conjecture, we argue that the presence of a single macroscopic cluster is an intermediate stage of a complex nucleation process: in smaller systems, this cluster is destabilized while larger systems form multiple clusters. Both limits lead to exponential cluster size distributions, controlled by very different length scales.
Application of classical non-linear Liouville dynamic approximations
Harter, Terry Lee (Virginia Polytechnic Institute and State University, 1988)
This dissertation examines the application of the Liouville operator to problems in classical mechanics. An approximation scheme or methodology is sought that would allow the calculation of the position and momentum of an object at a specified later time, given the initial values of the object's position and momentum at some specified earlier time. The approximation scheme utilizes matrix techniques to represent the Liouville operator. An approximation scheme using the Liouville operator is formulated and applied to several simple one-dimensional physical problems, whose solution is obtainable in terms of known analytic functions. The scheme is shown to be extendable relative to cross products and powers of the variables involved. The approximation scheme is applied to a more complicated one-dimensional problem, a quartic perturbed simple harmonic oscillator, whose solution is not capable of being expressed in terms of simple analytic functions. Data produced by the application of the approximation scheme to the perturbed quartic harmonic oscillator is analyzed statistically and graphically. The scheme is reapplied to the solution of the same problem with the incorporation of a drag term, and the results analyzed. The scheme is then applied to a simple physical pendulum having a functionalized potential in order to ascertain the limits of the approximation technique. The approximation scheme is next applied to a two-dimensional non-perturbed Kepler problem. The data produced is analyzed statistically and graphically. Conclusions are drawn and suggestions are made in order to continue the research in several of the areas presented.
Bistability in an Ising model with non-Hamiltonian dynamics
Heringa, J. R.; Shinkai, H.; Blote, H. W. J.; Hoogland, A.; Zia, Royce K. P. (American Physical Society, 1992-03)
We investigate the phenomenon of magnetization bistability in a two-dimensional Ising model with a non-Hamiltonian Glauber dynamics by means of Monte Carlo simulations. This effect has previously been observed in the Toom model, which supports two stable phases with different magnetizations, even in the presence of a nonzero field. We find that such bistability is also present in an Ising model in which the transition probabilities are expressed in terms of Boltzmann factors depending only on the nearest-neighbor spins and the associated bond strengths. The strength on each bond assumes different values with respect to the spins at either of its ends, introducing an asymmetry like that of the Toom model.
Classical and quantum gravity with Ashtekar variables
Soo, Chopin (Virginia Tech, 1992)
This thesis is a study of classical and quantum gravity with Ashtekar variables. The Ashtekar constraints are shown to capture the essence of the constraints and constraint algebra of General Relativity in four dimensions. A classification scheme of the solution space of the Ashtekar constraints is proposed and the corresponding physics is investigated. The manifestly covariant equations of motion for the Ashtekar variables are derived. Explicit examples are discussed and new classical solutions of General Relativity are constructed by exploiting the properties of the Ashtekar variables. Non-perturbative canonical quantization of the theory is performed. The ordering of the quantum constraints as well as the formal closure of the quantum constraint algebra are explored. A detailed Becchi-Rouet-Stora-Tyutin (BRST) analysis of the theory is given. The results demonstrate explicitly that in quantum gravity, fluctuations in topology can occur and there are strong evidences of phases in the theory. There is a phase which is described by a topological quantum field theory (TQFT) of the Donaldson-Witten type and an Abelian antiinstanton phase wherein self-interactions of the gravitational fields produce symmetry breaking from SO(3) to U(1). The full theory is much richer and includes fluctuations which bring the system out of the various restricted sectors while preserving diffeomorphism invariance. Invariants of the quantum theory with are constructed through BRST descents. They provide a clear and systematic characterization of non-local observables in quantum gravity, and can yield further differential invariants of four-manifolds.
Coarsening of "clouds" and dynamic scaling in a far-from-equilibrium model system
Adams, D. A.; Schmittmann, Beate; Zia, Royce K. P. (American Physical Society, 2007-04)
A two-dimensional lattice gas of two species, driven in opposite directions by an external force, undergoes a jamming transition if the filling fraction is sufficiently high. Using Monte Carlo simulations, we investigate the growth of these jams ('' clouds ''), as the system approaches a nonequilibrium steady state from a disordered initial state. We monitor the dynamic structure factor S(k(x),k(y);t) and find that the k(x)=0 component exhibits dynamic scaling, of the form S(0,k(y);t)=t(beta)S(k(y)t(alpha)). Over a significant range of times, we observe excellent data collapse with alpha=1/2 and beta=1. The effects of varying filling fraction and driving force are discussed.
Competition between multiple totally asymmetric simple exclusion processes for a finite pool of resources
Cook, L. J.; Zia, Royce K. P.; Schmittmann, Beate (American Physical Society, 2009-09)
Using Monte Carlo simulations and a domain-wall theory, we discuss the effect of coupling several totally asymmetric simple exclusion processes (TASEPs) to a finite reservoir of particles. This simple model mimics directed biological transport processes in the presence of finite resources such as protein synthesis limited by a finite pool of ribosomes. If all TASEPs have equal length, we find behavior which is analogous to a single TASEP coupled to a finite pool. For the more generic case of chains with different lengths, several unanticipated regimes emerge. A generalized domain-wall theory captures our findings in good agreement with simulation results.
Composite models of quarks and leptons
Geng, Chaqiang (Virginia Polytechnic Institute and State University, 1987)
We review the various constraints on composite models of quarks and leptons. Some dynamical mechanisms for chiral symmetry breaking in chiral preon models are discussed. We have constructed several "realistic candidate" chiral preon models satisfying complementarity between the Higgs and confining phases. The models predict three to four generations of ordinary quarks and leptons.
Cooperative Behavior in Driven Lattice Systems with Shifted Periodic Boundary Conditions
Anderson, Mark Jule Jr. (Virginia Tech, 1998-04-17)
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.
Critical behaviour of driven bilayer systems: a field-theoretic renormalization group study
Täuber, Uwe C.; Schmittmann, Beate; Zia, Royce K. P. (IOP, 2001-10-26)
Crossover of interfacial dynamics
Jasnow, D.; Zia, Royce K. P. (American Physical Society, 1987-09)
Some dynamical models in which there is a significant interplay between interfacial and bulk degrees of freedom are treated at the mean-field (or Van Hove) level from a coarse-grained microscopic viewpoint. Specifically, the near-equilibrium interfacial dynamics of two of the simplest models with conservation laws, models B and C, are studied with use of (as appropriate) variational techniques, perturbation theory, and (with certain additional simplifications) exact solutions. Use of these methods allows the dispersion relation for interfacial modes to be interpolated between the ‘‘hydrodynamical’’ and critical regimes. The crossover scaling behavior lends support to renormalization-group methods near d=1 which focus on the interfacial modes but nonetheless extract the bulk dynamical exponent as well as the crossover.
The diagramatical solution of the two-impurity Kondo problem
Zhou, Chen (Virginia Polytechnic Institute and State University, 1988)
The problem of the two-impurity Kondo problem is studied via the perturbative diagrammatical method. The high-temperature magnetic susceptibility is calculated to fourth order in the coupling constant J for different regimes. The integral equations for the ground state energy are established and solved numerically. The two-stage Kondo effect and corresponding energy scales are found which agree with the scaling results.
Dynamics of Competition using a Bit String Model with Age Structure and Mutations
Astalos, Robert Joseph (Virginia Tech, 2001-04-17)
Using Monte Carlo simulations and analytic methods, we examine the dynamics of inter-species competition using the Penna bit-string model. We begin with a study of the steady state with a single species, then proceed to the dynamics of competition between two species. When the species are not evenly matched in fitness, a simple differential equation provides a satisfactory model of the behavior of the system. However, when the species are equally fit, we show that a model, originally proposed to describe population genetics [Fisher,Wright], is required. When mutations are allowed between the competing species, the dynamics becomes more interesting. The mutation rate becomes a parameter that dictates the steady state behavior. If the two species are not equally fit, the value of the mutation rate determines whether the longer-lived or faster reproducing species is favored. With two species that are equally fit, the steady state varies with mutation rate from a single peaked to a double peaked distribution. This behavior is shown to be well described by an extension to the Fisher-Wright model mentioned above. Finally, we describe the preliminary results of a few new lines of investigation, and suggest ideas for further study of the dynamics of this model.
Effects of anisotropic surface tension on first-order-transition singularities
Zia, Royce K. P.; Wallace, D. J. (American Physical Society, 1985-02)
For systems displaying two-phase coexistence without rotational invariance, anisotropic surface tension and nonspherical droplets are present. To study small fluctuations around such droplets, we construct a natural coordinate system and find the quadratic form. In general, their spectrum differs from the isotropic case and affects the nature of the first-order transition singularities. However, in two bulk dimensions, the spectrum is sufficiently simple that the singularity is universal.
Effects of gravity on equilibrium crystal shapes
Gittis, Apostolos Georgios (Virginia Polytechnic Institute and State University, 1988)
The effects of gravity on the two-dimensional equilibrium shapes (ES) of crystals and menisci are investigated for different geometries (positions) of the substrate. In the gravity-free case, the equilibrium crystal shape (ECS) is characterized by a scale invariance. The presence of gravity breaks the scale invariance and the resulting ECS changes as the volume of the crystal V is changed. Moreover, the presence of gravity breaks the translational invariance along the direction it acts. Physically realized by the necessity of a support, this is manifested by the existence of an inhomogeneous effective pressure P_eff, which divides the space into two regions, with P_eff either negative or positive. The ECS changes as the crystal passes from one region to another, being concave where P_eff < 0, and convex where P_eff > 0. In all cases it was possible to express the corresponding ECS in terms of the gravity-free one. For the hung crystal, i.e., a crystal pinned to a vertical wall at the top, it is shown that some orientations are missing from the ECS that otherwise will be present in the gravity-free ECS, adsorbed on the same substrate. Thus, facets could disappear from the crystal shape as the volume V or the gravitational acceleration g is increased. A critical volume V_c is found, so that if the crystal volume V exceeds V_c, the crystal cannot be pinned. The ECS can exhibit both concave and convex portions. For a crystal, pinned to a vertical wall at its lower end, we find that it will never develop a concave part. On the other hand, new orientations, absent from the gravity-free crystal, will be present on its ECS. The ES of a free and pinned crystal meniscus is also solved and an expression for the excess (depleted) volume AV is derived. The solution for the crystal meniscus between two walls is also presented. For the pendant crystal, i.e., a crystal hanging from a horizontal support, we find that it can exhibit both concave and convex portions on its ECS. When it develops a concave part, new orientations will appear, compared to the gravity-free case. An intuitive stability criterion is introduced, according to which only crystals wetting the substrate can develop a concave portion before they break. The treatment of a crystal on an inclined substrate shows the complications that arise in determining the ES for a general position of the support as a result of the conflict between the directions associated with gravity and support. An expression for the facet length in the presence of gravity is obtained that is valid for all types of support. For crystal shapes that display a concave portion it offers a very convenient way to experimentally measure step free energies. Thus, by breaking scale invariance, the presence of gravity allows absolute measures of surface energy in contrast to the gravity-free case, where the facet length is proportional to the step free energy by an unknown scale.
Effects of gravity on equilibrium crystal shapes: Droplets hung on a wall
Zia, Royce K. P.; Gittis, A. (American Physical Society, 1987-04)
General properties of equilibrium crystal shapes pinned on a vertical wall and subject to gravity are sought. For two-dimensional crystals, or three-dimensional ones with axial symmetry held in suitable geometries, we are able to express the results in terms of the well-known gravity-free Wulff-Winterbottom shapes. All results are valid for an arbitrary, given, orientation-dependent surface-tension function.
Effects of next-nearest-neighbor interactions on the orientation dependence of step stiffness: Reconciling theory with experiment for Cu(001)
Stasevich, T. J.; Einstein, T. L.; Zia, Royce K. P.; Giesen, M.; Ibach, H.; Szalma, F. (American Physical Society, 2004-12)
Within the solid-on-solid (SOS) approximation, we carry out a calculation of the orientational dependence of the step stiffness on a square lattice with nearest- and next-nearest-neighbor interactions. At low temperature our result reduces to a simple, transparent expression. The effect of the strongest trio (three-site, nonpairwise) interaction can easily be incorporated by modifying the interpretation of the two pairwise energies. The work is motivated by a calculation based on nearest neighbors that underestimates the stiffness by a factor of 4 in directions away from close-packed directions, and a subsequent estimate of the stiffness in the two high-symmetry directions alone that suggested that inclusion of next-nearest-neighbor attractions could fully explain the discrepancy. As in these earlier papers, the discussion focuses on Cu(001).
Efficient Numeric Computation of a Phase Diagram in Biased Diffusion of Two Species
Parks, Michael Lawrence (Virginia Tech, 2000-05-09)
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.
Entrainment and Unit Velocity: Surprises in an Accelerated Exclusion Process
Dong, J. J.; Klumpp, S.; Zia, Royce K. P. (American Physical Society, 2012-09-27)
We introduce a class of distance-dependent interactions in an accelerated exclusion process inspired by the observation of transcribing RNA polymerase speeding up when "pushed" by a trailing one. On a ring, the accelerated exclusion process steady state displays a discontinuous transition, from being homogeneous (with augmented currents) to phase segregated. In the latter state, the holes appear loosely bound and move together, much like a train. Surprisingly, the current-density relation is simply J = 1 - rho, signifying that the "hole train" travels with unit velocity.
Epidemic Spreading on Preferred Degree Adaptive Networks
Jolad, Shivakumar; Liu, Wenjia; Schmittmann, Beate; Zia, Royce K. P. (PLOS, 2012-11-26)
We study the standard SIS model of epidemic spreading on networks where individuals have a fluctuating number of connections around a preferred degree. Using very simple rules for forming such preferred degree networks, we find some unusual statistical properties not found in familiar Erdös-Rényi or scale free networks. By letting depend on the fraction of infected individuals, we model the behavioral changes in response to how the extent of the epidemic is perceived. In our models, the behavioral adaptations can be either ‘blind’ or ‘selective’ – depending on whether a node adapts by cutting or adding links to randomly chosen partners or selectively, based on the state of the partner. For a frozen preferred network, we find that the infection threshold follows the heterogeneous mean field result and the phase diagram matches the predictions of the annealed adjacency matrix (AAM) approach. With ‘blind’ adaptations, although the epidemic threshold remains unchanged, the infection level is substantially affected, depending on the details of the adaptation. The ‘selective’ adaptive SIS models are most interesting. Both the threshold and the level of infection changes, controlled not only by how the adaptations are implemented but also how often the nodes cut/add links (compared to the time scales of the epidemic spreading). A simple mean field theory is presented for the selective adaptations which capture the qualitative and some of the quantitative features of the infection phase diagram.

Browsing by Author "Zia, Royce K. P."

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