Browsing by Author "Zia, Royce K. P."
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- Analytic Results for Hopping Models with Excluded Volume ConstraintToroczkai, 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 equilibriumGeorgiev, 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.
- Bistability in an Ising model with non-Hamiltonian dynamicsHeringa, 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.
- Coarsening of "clouds" and dynamic scaling in a far-from-equilibrium model systemAdams, 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 resourcesCook, 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.
- Cooperative Behavior in Driven Lattice Systems with Shifted Periodic Boundary ConditionsAnderson, 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 studyTäuber, Uwe C.; Schmittmann, Beate; Zia, Royce K. P. (IOP, 2001-10-26)
- Crossover of interfacial dynamicsJasnow, 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.
- Dynamics of Competition using a Bit String Model with Age Structure and MutationsAstalos, 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 singularitiesZia, 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: Droplets hung on a wallZia, 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 SpeciesParks, 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 ProcessDong, 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 NetworksJolad, 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.
- Equilibrium budding and vesiculation in the curvature model of fluid lipid vesiclesMiao, L.; Fourcade, B.; Rao, M. D.; Wortis, M.; Zia, Royce K. P. (American Physical Society, 1991-06)According to a model introduced by Helfrich [Z. Naturforsch. 28c, 693 (1973)], the shape of a closed lipid vesicle is determined by minimization of the total bending energy at fixed surface area and enclosed volume. We show that, in the appropriate regime, this model predicts both budding (the eruption of a satellite connected to the parent volume via a neck) and vesiculation (the special case when the neck radius goes to zero). Vesiculation occurs when the minimum is located at a boundary in the space of configurations. Successive vesiculations produce multiplets, in which the minimum-energy configuration consists of several bodies coexisting through infinitesimal necks. We study the sequence of shapes and shape transitions followed by a spherical vesicle of radius R(v), large on the scale R0 set by the spontaneous curvature, as its area A increases at constant volume V = 4-pi-R(v)3/3. Such a vesicle periodically sheds excess area into a set of smaller spheres with radii comparable to R0. We map out this (shape) phase diagram at large volume. In this region the phase diagram is dominated by multiples and reflects the details of the shedding process. The overall effect of successive vesiculations is to reduce the energy from a quantity of order R(v)2 down to zero or near zero when the area reaches 3V/R0; however, the decrease is not uniform and the energy E(A,V) is not convex.
- Exact dynamics of a reaction-diffusion model with spatially alternating ratesMobilia, M.; Schmittmann, Beate; Zia, Royce K. P. (American Physical Society, 2005-05)We present the exact solution for the full dynamics of a nonequilibrium spin chain and its dual reaction-diffusion model, for arbitrary initial conditions. The spin chain is driven out of equilibrium by coupling alternating spins to two thermal baths at different temperatures. In the reaction-diffusion model, this translates into spatially alternating rates for particle creation and annihilation, and even negative "temperatures" have a perfectly natural interpretation. Observables of interest include the magnetization, the particle density, and all correlation functions for both models. Two generic types of time dependence are found: if both temperatures are positive, the magnetization, density, and correlation functions decay exponentially to their steady-state values. In contrast, if one of the temperatures is negative, damped oscillations are observed in all quantities. They can be traced to a subtle competition of pair creation and annihilation on the two sublattices. We comment on the limitations of mean-field theory and propose an experimental realization of our model in certain conjugated polymers and linear chain compounds.
- An Experimental Spatio-Temporal Analysis of Separated Flows Over Bluff Bodies Using Quantitative Flow VisualizationVlachos, Pavlos P. (Virginia Tech, 2000-05-11)In order to study three-dimensional unsteady turbulent flow fields such as the wakes of bluff bodies, a Digital Particle Image Velocimetry (DPIV) system was developed. This system allows non-intrusive two-dimensional and time varying velocity measurements. Software and hardware modifications necessary to enhance the capabilities of the system were preformed, resulting in increased frequency resolution. However, due to hardware limitations and limitations inherited from the implementation of the method, space resolution is reduced. Subsequently, digital image processing tools to improve the space resolutions were developed. The advantages and limitations of the method for the study of turbulent flows are presented in detail. The developed system is employed in the documentation of time-varying turbulent flow fields. Initially we study the spanwise variation of the near wake of a low-aspect ratio, surface-mounted, circular cylinder piercing a free surface. The asymmetry of the end conditions combined with the natural unsteadiness of the vortex shedding generates a very complex flow filed which is difficult to study with conventional methods. By employing the aforementioned system we are able to reveal a departure of the two-dimensional character of the flow in the form of oblique vortex shedding. The effect of free surface on the vortex formation length and on the vortex reconnection process is documented. Near the free surface the alternate mode of vortex shedding is suppressed, leading to simultaneous shedding of vortices in the wake. Indications of vortex dislocations and change of the vortex axis in order to reconnect to the free surface are observed. Finally, a novel approach of reconstructing the three-dimensional, time -varying volume of the flow field by obtaining simultaneous measurements of Laser Doppler Velocimetry and Particle Image Velocimetry planes is presented. The same field is investigated with focus on the streamwise structures. Three-dimensional streamwise vortical structures are known to exist due to instabilities of plane shear layers. Similar streamwise vortices, also known as braid vortices have been observed in the past in the wake of circular cylinders with symmetric boundary conditions. The present spatio-temporal analysis demonstrated coexistence of two types of streamwise vortices in the wake, bilge and braid type of vortices. These may be due to the three dimensionality introduced by the free surface. In addition, the sufficient time resolution allowed the detection of the primary Von-Karman vortex through a plane of interrogation normal to the free stream, thus revealing the spanwise variation of the vortex shedding and its evolution at different downstream stations. The combination of the effect of the asymmetric boundary conditions with a free surface is investigated by adding one more source of three-dimensionality in terms of inclination of the cylinder axis. Hydrogen-bubble and particle-flow visualizations are preformed in combination with Laser-Doppler Velocimetry measurements. From both qualitative and quantitative results the effects of inclination and Froude number are documented. It is proved that the vortex shedding is suppressed for high values of the Froude number, however the inclination counteracts the vortex suppression and favors the vortex shedding mechanism. In addition, in the region of the no-slip boundary condition the flow is dominated by the effect of the horseshoe vortex. The case of a three-dimensional separated flow over a surface-mounted prism is investigated using a modified version of the system. The character of the separated from the leading edge corner shear layer and the formed separation bubble are documented in space and time along the mid-plane of symmetry of the body. Three different flows corresponding to different Reynolds numbers are studied. The unsteadiness of the flow is presented indicating a pseudo-periodic character. Large-scale, low-frequency oscillations of the shear layer that have been observed in the past using point measurement methods are now confirmed by means of a whole field velocity measurement, technique allowing a holistic view of the flow. In addition, the unsteadiness of the point of reattachment is associated with the flapping of the shear layer and the shedding of vorticity in the wake. Finally, it is demonstrated that the apparent vortex shedding mechanism of such flows is dependent on the interaction of the primary vortex of the separation bubble with a secondary vortex formed by the separation of the reverse flow boundary layer. By performing measurements with such time and space resolution the inadequacy of time averaged or point measurement methods for the treatment of such complex and unsteady flow fields becomes evident. In final case we employ Particle-Image Velocimetry to show the effect of unsteady excitation on two-dimensional separated flow over a sharp edged airfoil. It is proved that such an approach can be used to effectively control and organize the character of the flow, potentially leading to lift increase and drug reduction of bluff bodies
- Finger formation in a driven diffusive systemBoal, D. H.; Schmittmann, Beate; Zia, Royce K. P. (American Physical Society, 1991-05)A driven diffusive lattice gas is studied in a rectangular geometry: particles are fed in at one side and extracted at the other, after being swept through the system by a uniform driving field. Being periodic in the transverse direction, the lattice lies on the surface of a cylinder. The resulting nonequilibrium steady state depends strongly on this choice of boundary conditions. Both Monte Carlo and analytic techniques are employed to investigate the structure of typical configurations, the density profile, the steady-state current, and the nearest-neighbor correlations. As the temperature is lowered in a finite system, the simulations indicate a crossover from a disordered to an ordered state that is characterized by a backgammonlike pattern of alternating high- and low-density regions ("fingers"). For fixed strengths of the field and interparticle attraction, the average number of fingers is controlled by the ratio of the transverse to the longitudinal system size. Whether the crossover corresponds to an actual phase transition, where typical thermodynamic observables become singular, remains to be determined.
- Fluid-magnet universality: Renormalization-group analysis of φ5 operatorsNicoll, J. F.; Zia, Royce K. P. (American Physical Society, 1981-06)The question of a possible difference between the universality classes of fluids and Ising-like magnets is addressed by perturbation theory and the renormalization group. The most dangerous possibility is that of an φ5 addition to the usual φ4 theory. We show that no φ5 fixed point exists in the framework of an expansion around d=10/3. Further we show that to O(ε2/4), ε4≡4−d, the ordinary φ4 fixed point is stable against the perturbations that mix with φ5. Two new correction-to-scaling exponents are found. One of the exponents, Δ5, is poorly determined with a range of values from 0.5 to 1.0 compatible with the O(ε2/4) result. However, its positivity rules out a separate fluid fixed point, indicating fluid-magnet asymptotic universality. The second exponent, Δ3, can be determined exactly: Δ3=1−α−β. This implies the universal existence of a contribution to the fluid diameter scaling like the internal energy.
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