Scholarly Works, Center for Stochastic Processes in Science and Engineering (CSPISE)

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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.
Bound states of hydrogen atom in a theory with minimal length uncertainty relations
Slawny, Joseph A. (AIP Publishing, 2007-05)
The following properties of bound states of hydrogen atom in a theory with noncommuting position operators are investigated: their number, multiplicities, accidental degeneracy, localization, and dependence on the values of deformation parameters. (c) 2007 American Institute of Physics.
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.
Driven diffusive systems: How steady states depend on dynamics
Kwak, W.; Landau, D. P.; Schmittmann, Beate (American Physical Society, 2004-06)
In contrast to equilibrium systems, nonequilibrium steady states depend explicitly on the underlying dynamics. Using Monte Carlo simulations with Metropolis, Glauber, and heat bath rates, we illustrate this expectation for an Ising lattice gas, driven far from equilibrium by an "electric" field. While heat bath and Glauber rates generate essentially identical data for structure factors and two-point correlations, Metropolis rates give noticeably weaker correlations, as if the "effective" temperature were higher in the latter case. We also measure energy histograms and define a simple ratio which is exactly known and closely related to the Boltzmann factor for the equilibrium case. For the driven system, the ratio probes a thermodynamic derivative which is found to be dependent on dynamics.
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 receptor clustering on ligand dissociation kinetics: Theory and simulations
Gopalakrishnan, Mahima; Forsten-Williams, Kimberly; Nugent, Matthew A.; Täuber, Uwe C. (Cell Press, 2005-12-01)
Receptor-ligand binding is a critical first step in signal transduction and the duration of the interaction can impact signal generation. In mammalian cells, clustering of receptors may be facilitated by heterogeneous zones of lipids, known as lipid rafts. In vitro experiments show that disruption of rafts significantly alters the dissociation of fibrbroblast growth factor-2 (FGF2) from heparan sulfate proteoglycans (HSPGs), co-receptors for FGF-2. In this article, we develop a continuum stochastic formalism to address how receptor clustering might influence ligand rebinding. We find that clusters reduce the effective dissociation rate dramatically when the clusters are dense and the overall surface density of receptors is low. The effect is much less pronounced in the case of high receptor density and shows nonmonotonic behavior with time. These predictions are verified via lattice Monte Carlo simulations. Comparison with FGF-2-HSPG experimental results is made and suggests that the theory could be used to analyze similar biological systems. We further present an analysis of an additional cooperative internal-diffusion model that might be used by other systems to increase ligand retention when simple rebinding is insufficient.
Equilibrium budding and vesiculation in the curvature model of fluid lipid vesicles
Miao, 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 rates
Mobilia, 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.
Field-induced vacancy localization in a driven lattice gas: Scaling of steady states
Thies, M.; Schmittmann, Beate (American Physical Society, 2000-01)
With the help of Monte Carlo simulations and a mean-field theory, we investigate the ordered steady-state structures resulting from the motion of a single vacancy on a periodic lattice which is filled with two species of oppositely "charged" particles. An external field biases particle-vacancy exchanges according to the particle's charge, subject to an excluded volume constraint. The steady state exhibits charge segregation, and the vacancy is localized at one of the two characteristic interfaces. Charge and hole density profiles, an appropriate order parameter, and the interfacial regions themselves exhibit characteristic scaling properties with system size and field strength. The lattice spacing is found to play a significant role within the mean-field theory.
Finger formation in a driven diffusive system
Boal, 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.
Inhomogeneous exclusion processes with extended objects: The effect of defect locations
Dong, J. J.; Schmittmann, Beate; Zia, Royce K. P. (American Physical Society, 2007-11)
We study the effects of local inhomogeneities, i.e., slow sites of hopping rate q < 1, in a totally asymmetric simple exclusion process for particles of size l >= 1 (in units of the lattice spacing). We compare the simulation results of l=1 and l>1 and notice that the existence of local defects has qualitatively similar effects on the steady state. We focus on the stationary current as well as the density profiles. If there is only a single slow site in the system, we observe a significant dependence of the current on the location of the slow site for both l=1 and l>1 cases. When two slow sites are introduced, more intriguing phenomena emerge, e.g., dramatic decreases in the current when the two are close together. In addition, we study the asymptotic behavior when q -> 0. We also explore the associated density profiles and compare our findings to an earlier study using a simple mean-field theory. We then outline the biological significance of these effects.
New phase in the one-dimensional t-J model
Chen, Y. C.; Lee, T. K. (American Physical Society, 1993-05)
A new phase of a gas of pairs of electrons bounded in a singlet state is found in the one-dimensional t -J model for J > 2t and the density of electrons less than 0.2. This phase was conjectured in the study of the diagonalization of small lattices [Phys. Rev. Lett. 66, 2388 (1991)]. The existence of this new phase for much larger lattice sizes is demonstrated by a combination of two numerical methods, the variational Monte Carlo and the power method. A trial wave function for this phase is proposed and shown to be in good agreement with the ground state obtained by the power method.
Position and length operators in a theory with minimal length
Slawny, Joseph A. (AIP Publishing, 2007-05)
Relations between the notions of fundamental and minimal lengths, and duality, in a system with minimal length uncertainty relations are examined. Self-adjoint versions of operators relevant to the problem, and their spectra, are analyzed in detail. (c) 2007 American Institute of Physics.
Roughness, spatial, and temporal correlations of an interface in a driven nonequilibrium lattice gas
Leung, K. T.; Mon, K. K.; Valles, J. L.; Zia, Royce K. P. (American Physical Society, 1989-05)
The interface of a stochastic Ising lattice gas driven into a nonequilibrium steady state by a constant, uniform electric field E parallel to the interface is studied by extensive Monte Carlo simulation in two dimensions. Dependence on the system size and the field strength of the interface profile, roughness, time, and spatial correlation functions and structure factors are found numerically, by means of a coarse-graining method. The interface at zero field is shown to be rough by the divergence of both the interface width and correlation time. As soon as E is turned on, the interface becomes smooth. We argue that the general results may be extended to other similar nonequilibrium systems, and in higher dimensions.
Spin-charge separation in the two-dimensional Hubbard and t-J models at low electronic density
Chen, Y. C.; Moreo, A.; Ortolani, F.; Dagotto, E.; Lee, T. K. (American Physical Society, 1994-07)
The spin- and density-correlation functions of the two-dimensional Hubbard model at low electronic density [n] are calculated in the ground state by using the power method, and at finite temperatures by using the quantum Monte Carlo technique. Both approaches produce similar results, which are in close agreement with numerical and high-temperature-expansion results for the two-dimensional t-J model. Using perturbative approximations, we show that the examination of the density-correlation function alone is not enough to support recent claims in the literature that suggested spin and charge separation in the low electronic density regime of the t-J model.
Stationary correlations for a far-from-equilibrium spin chain
Schmittmann, Beate; Schmuser, F. (American Physical Society, 2002-10)
A kinetic one-dimensional Ising model on a ring evolves according to a generalization of Glauber rates, such that spins at even (odd) lattice sites experience a temperature T(e) (T(o)). Detailed balance is violated so that the spin chain settles into a nonequilibrium stationary state, characterized by multiple interactions of increasing range and spin order. We derive the equations of motion for arbitrary correlation functions and solve them to obtain an exact representation of the steady state. Two nontrivial amplitudes reflect the sublattice symmetries; otherwise, correlations decay exponentially, modulo the periodicity of the ring. In the long-chain limit, they factorize into products of two-point functions, in precise analogy to the equilibrium Ising chain. The exact solution confirms the expectation, based on simulations and renormalization group arguments, that the long-time, long-distance behavior of this two-temperature model is Ising-like, in spite of the apparent complexity of the stationary distribution.
Steady states of a nonequilibrium lattice gas
Lyman, E.; Schmittmann, Beate (American Physical Society, 2005-09)
We present a Monte Carlo study of a lattice gas driven out of equilibrium by a local hopping bias. Sites can be empty or occupied by one of two types of particles, which are distinguished by their response to the hopping bias. All particles interact via excluded volume and a nearest-neighbor attractive force. The main result is a phase diagram with three phases: a homogeneous phase and two distinct ordered phases. Continuous boundaries separate the homogeneous phase from the ordered phases, and a first-order line separates the two ordered phases. The three lines merge in a nonequilibrium bicritical point.
Strongly anisotropic roughness in surfaces driven by an oblique particle flux
Schmittmann, Beate; Pruessner, G.; Janssen, H. K. (American Physical Society, 2006-05)
Using field theoretic renormalization, an MBE-type growth process with an obliquely incident influx of atoms is examined. The projection of the beam on the substrate plane selects a "parallel" direction, with rotational invariance restricted to the transverse directions. Depending on the behavior of an effective anisotropic surface tension, a line of second-order transitions is identified, as well as a line of potentially first-order transitions, joined by a multicritical point. Near the second-order transitions and the multicritical point, the surface roughness is strongly anisotropic. Four different roughness exponents are introduced and computed, describing the surface in different directions, in real or momentum space. The results presented challenge an earlier study of the multicritical point.
t-J model studied by the power Lanczos method
Chen, Y. C.; Lee, T. K. (American Physical Society, 1995-03)
The initial trial wave function used in a simple ground-state projection method, the power method, is systematically improved by using Lanczos algorithm. Much faster convergence to the ground state achieved by using these wave functions significantly reduces the effect of the fermion sign problem. The results for the ground state of the two-dimensional t-J model are presented. The density correlation function for the t-J model at small J shows a surprisingly good agreement with that of a system of noninteracting hard-core bosons.
Transmutation of the vicinal surface exponent due to gravity
Avron, J. E.; Zia, Royce K. P. (American Physical Society, 1988-04)
Near the edge of a facet, vicinal surfaces curve away from the plane with exponent (3/2. With gravity normal to the facet the exponent may change to 3. Generally, there will be a crossover from 3 to (3/2 as the facet edge is approached. Comparison with experiments is made.

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