Showing papers in "Physical Review E in 1996"

Lattice Boltzmann simulations of liquid-gas and binary fluid systems

Abstract: We present the details of a lattice Boltzmann approach to phase separation in nonideal one- and two-component fluids. The collision rules are chosen such that the equilibrium state corresponds to an input free energy and the bulk flow is governed by the continuity, Navier-Stokes, and, for the binary fluid, a convection-diffusion equation. Numerical results are compared to simple analytic predictions to confirm that the equilibrium state is indeed thermodynamically consistent and that the kinetics of the approach to equilibrium lie within the expected universality classes. The approach is compared to other lattice Boltzmann simulations of nonideal systems. \textcopyright{} 1996 The American Physical Society.

Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method

Abstract: We describe an implementation of the recently proposed lattice Boltzmann based model of Shan and Chen [Phys. Rev. E 47, 1815 (1993); 49, 2941 (1994)] to simulate multicomponent flow in complex three-dimensional geometries such as porous media. The above method allows for the direct incorporation of fluid-fluid and fluid-solid interactions as well as an applied external force. As a test of this method, we obtained Poiseuille flow for the case of a single fluid driven by a constant body force and obtained results consistent with Laplace's law for the case of two immiscible fluids. The displacement of one fluid by another in a porous medium was then modeled. The relative permeability for different wetting fluid saturations of a microtomography-generated image of sandstone was calculated and compared favorably with experiment. In addition, we show that a first-order phase transition, in three dimensions, may be obtained by this lattice Boltzman method, demonstrating the potential for modeling phase transitions and multiphase flow in porous media. \textcopyright{} 1996 The American Physical Society.

Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support.

Abstract: A simple scaling argument shows that most integrable evolutionary systems, which are known to admit a bi-Hamiltonian structure, are, in fact, governed by a compatible trio of Hamiltonian structures. We demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons (solitary-wave solutions having compact support), or cusped and/or peaked solitons. A general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified Korteweg--de Vries equation, the nonlinear Schr\"odinger equation, the integrable Boussinesq system used to model the two-way propagation of shallow water waves, and the Ito system of coupled nonlinear wave equations. These hierarchies include a remarkable variety of interesting integrable nonlinear differential equations. \textcopyright{} 1996 The American Physical Society.

Model for collisions in granular gases

Abstract: We propose a model for collisions between particles of a granular material and calculate the restitution coefficients for the normal and tangential motion as functions of the impact velocity from considerations of dissipative viscoelastic collisions. Existing models of impact with dissipation as well as the classical Hertz impact theory are included in the present model as special cases. We find that the type of collision (smooth, reflecting or sticky) is determined by the impact velocity and by the surface properties of the colliding grains. We observe a rather nontrivial dependence of the tangential restitution coefficient on the impact velocity. \textcopyright{} 1996 The American Physical Society.

Nonconservative lagrangian and Hamiltonian mechanics

Abstract: Traditional Lagrangian and Hamiltonian mechanics cannot be used with nonconservative forces such as friction. A method is proposed that uses a Lagrangian containing derivatives of fractional order. A direct calculation gives an Euler-Lagrange equation of motion for nonconservative forces. Conjugate momenta are defined and Hamilton's equations are derived using generalized classical mechanics with fractional and higher-order derivatives. The method is applied to the case of a classical frictional force proportional to velocity. \textcopyright{} 1996 The American Physical Society.

Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics

Abstract: We present mathematical results which dramatically enhance the computational efficiency of the phase-field method for modeling the solidification of a pure material. These results make it possible to resolve a smaller capillary length to interface thickness ratio and thus render smaller undercooling and three-dimensional computations accessible. Furthermore, they allow one to choose computational parameters to produce a Gibbs-Thomson condition with an arbitrary kinetic coefficient. The method is tested for dendritic growth in two dimensions with zero kinetic coefficient. Simulations yield dendrites with tip velocities and tip shapes which agree within a few percent with numerical Green's function solutions of the steady-state growth problem.

Generalized synchronization of chaos: The auxiliary system approach.

Abstract: Synchronization of chaotic oscillators in a generalized sense leads to richer behavior than identical chaotic oscillations in coupled systems. It may imply a more complicated connection between the synchronized trajectories in the state spaces of coupled systems. We suggest a method here that can be used to detect and study generalized synchronization in drive-response systems. This technique, the auxiliary system method, utilizes a second, identical response system to monitor the synchronized motions. The method can be implemented both numerically and experimentally and in some cases it leads to analytical results for generalized synchronization.

Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral.

Abstract: A numerical simulation algorithm that is exact for any time step \ensuremath{\Delta}tg0 is derived for the Ornstein-Uhlenbeck process X(t) and its time integral Y(t). The algorithm allows one to make efficient, unapproximated simulations of, for instance, the velocity and position components of a particle undergoing Brownian motion, and the electric current and transported charge in a simple R-L circuit, provided appropriate values are assigned to the Ornstein-Uhlenbeck relaxation time \ensuremath{\tau} and diffusion constant c. A simple Taylor expansion in \ensuremath{\Delta}t of the exact simulation formulas shows how the first-order simulation formulas, which are implicit in the Langevin equation for X(t) and the defining equation for Y(t), are modified in second order. The exact simulation algorithm is used here to illustrate the zero-\ensuremath{\tau} limit theorem. \textcopyright{} 1996 The American Physical Society.

Lacunarity analysis: A general technique for the analysis of spatial patterns

Abstract: Lacunarity analysis is a multiscaled method for describing patterns of spatial dispersion. It can be used with both binary and quantitative data in one, two, and three dimensions. Although originally developed for fractal objects, the method is more general and can be readily used to describe nonfractal and multifractal patterns. Lacunarity analysis is broadly applicable to many data sets used in the natural sciences; we illustrate its application to both geological and ecological data. {copyright} {ital 1996 The American Physical Society.}

Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis

Abstract: Driven anomalous diffusions (such as those occurring in some surface growths) are currently described through the nonlinear Fokker-Planck-like equation $(\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}){p}^{\ensuremath{\mu}}=\ensuremath{-}(\frac{\ensuremath{\partial}}{\ensuremath{\partial}x})[F(x){p}^{\ensuremath{\mu}}]+D(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{x}^{2}}){p}^{\ensuremath{ u}}$ [$(\ensuremath{\mu}, \ensuremath{ u})\ensuremath{\in}{\mathcal{R}}^{2}$; $F(x)={k}_{1}\ensuremath{-}{k}_{2}x$ is the external force; ${k}_{2}g~0$]. We exhibit here the (physically relevant) exact solution for all $(x, t)$. This solution was found through an ansatz based on the generalized entropic form ${S}_{q}[p]=\frac{{1\ensuremath{-}\ensuremath{\int}\mathrm{du}{[p(u)]}^{q}}}{(q\ensuremath{-}1)}$ (with $q\ensuremath{\in}\mathcal{R}$), in a completely analogous manner through which the usual entropy ${S}_{1}[p]=\ensuremath{-}\ensuremath{\int}\mathrm{dup}(u)\mathrm{ln}p(u)$ is known to provide the correct ansatz for exactly solving the standard Fokker-Planck equation ($\ensuremath{\mu}=\ensuremath{ u}=1$). This remarkably simple unification of normal diffusion ($q=1$), superdiffusion ($qg1$) and subdiffusion ($ql1$) occurs with $q=1+\ensuremath{\mu}\ensuremath{-}\ensuremath{ u}$.

Singularity-free interpretation of the thermodynamics of supercooled water

Abstract: The pronounced increases in isothermal compressibility, isobaric heat capacity, and in the magnitude of the thermal expansion coefficient of liquid water upon supercooling have been interpreted either in terms of a continuous, retracing spinodal curve bounding the superheated, stretched, and supercooled states of liquid water, or in terms of a metastable, low-temperature critical point. Common to these two scenarios is the existence of singularities associated with diverging density fluctuations at low temperature. We show that the increase in compressibility upon lowering the temperature of a liquid that expands on cooling, like water, is not contingent on any singular behavior, but rather is a thermodynamic necessity. We perform a thermodynamic analysis for an anomalous liquid (i.e., one that expands when cooled) in the absence of a retracing spinodal and show that one may in general expect a locus of compressibility extrema in the anomalous regime. Our analysis suggests that the simplest interpretation of the behavior of supercooled water consistent with experimental observations is free of singularities. We then develop a waterlike lattice model that exhibits no singular behavior, while capturing qualitative aspects of the thermodynamics of water. {copyright} {ital 1996 The American Physical Society.}

Experimental properties of complexity in traffic flow

Abstract: Experimental investigations of a complexity in traffic flow are presented. It is shown that this complexity is linked to space-time transitions between three qualitative different kinds of traffic: "free" traffic flow, "synchronized" traffic flow, and traffic jams. Peculiarities of "synchronized" traffic flow and jams that are responsible for a complex behavior of traffic are found.

Experimental features and characteristics of traffic jams

Abstract: Based on experimental investigations of traffic on highways it is shown that traffic jams can move stable through a highway keeping their structure and characteristic parameters for a long time (at least for about 50 min, when the jams moved through the longest, 13.1 km, section of the investigated highways). The experimental features of an almost stationary moving jam have been found. An occurrence of complex space-time structures of traffic inside a wide traffic jam has been observed. \textcopyright{} 1996 The American Physical Society.

Light diffusion with gain and random lasers

Abstract: In this paper we present calculations on light diffusion with amplification that can explain previous experiments on the spontaneous emission from such a medium. Also we discuss the experimental considerations on realizing a medium that both multiply scatters and amplifies light. In an amplifying random medium different processes can occur. We argue that one can distinguish three regimes depending on the amount of scattering, and discuss these regimes in the context of random laser action. @S1063-651X~96!05109-4#

Avalanche dynamics in evolution, growth, and depinning models

Abstract: The dynamics of complex systems in nature often occurs in terms of punctuations, or avalanches, rather than following a smooth, gradual path. A comprehensive theory of avalanche dynamics in models of growth, interface depinning, and evolution is presented. Specifically, we include the Bak-Sneppen evolution model, the Sneppen interface depinning model, the Zaitsev flux creep model, invasion percolation, and several other depinning models into a unified treatment encompassing a large class of far from equilibrium processes. The formation of fractal structures, the appearance of 1/f noise, diffusion with anomalous Hurst exponents, L\'evy flights, and punctuated equilibria can all be related to the same underlying avalanche dynamics. This dynamics can be represented as a fractal in d spatial plus one temporal dimension. The complex state can be reached either by tuning a parameter, or it can be self-organized. We present two exact equations for the avalanche behavior in the latter case. (1) The slow approach to the critical attractor, i.e., the process of self-organization, is governed by a ``gap'' equation for the divergence of avalanche sizes. (2) The hierarchical structure of avalanches is described by an equation for the average number of sites covered by an avalanche. The exponent \ensuremath{\gamma} governing the approach to the critical state appears as a constant rather than as a critical exponent. In addition, the conservation of activity in the stationary state manifests itself through the superuniversal result \ensuremath{\eta}=0. The exponent \ensuremath{\pi} for the L\'evy flight jumps between subsequent active sites can be related to other critical exponents through a study of ``backward avalanches.'' We develop a scaling theory that relates many of the critical exponents in this broad category of extremal models, representing different universality classes, to two basic exponents characterizing the fractal attractor. The exact equations and the derived set of scaling relations are consistent with numerical simulations of the above mentioned models. \textcopyright{} 1996 The American Physical Society.

Effect of strongly favorable substrate interactions on the thermal properties of ultrathin polymer films.

Abstract: The thermal behavior of ultrathin films of poly-(2)-vinylpyridine spin-cast on acid-cleaned silicon oxide substrates is considered. The interaction between the polymer and the substrate is polar in nature and very favorable. As a means of examining the thermal properties of the films, x-ray reflectivity is used to measure the temperature dependence of the film thickness. This experimentally measured thickness-temperature data is used to determine transition temperatures and thermal expansivities. Significantly increased transition temperatures (20-50 \ifmmode^\circ\else\textdegree\fi{}C above the measured bulk glass transition temperature) are observed in ultrathin polymer films. The transition temperature increases with decreasing film thickness, while the degree of thermal expansion below the transition temperature decreases with decreasing film thickness. If one assumes that a region of reduced chain mobility exists near the solid substrate-polymer interface, an analysis of the measured thermal expansion behavior below the transition temperature indicates that the length scale of substrate interactions is on the order of the macromolecular size.

Particle hopping models and traffic flow theory

Abstract: This paper shows how particle hopping models fit into the context of traffic flow theory, that is, it shows connections between fluid-dynamical traffic flow models, which derive from the Navier-Stokes equations, and particle hopping models. In some cases, these connections are exact and have long been established, but have never been viewed in the context of traffic theory. In other cases, critical behavior of traffic jam clusters can be compared to instabilities in the partial differential equations. Finally, it is shown how all this leads to a consistent picture of traffic jam dynamics. In consequence, this paper starts building a foundation of a comprehensive dynamic traffic theory, where strengths and weaknesses of different models (fluid-dynamical, car-following, particle hopping) can be compared, and thus allowing to systematically chose the appropriate model for a given question.

Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory

Abstract: Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points, deriving both amplitude equations and the center manifold. @S1063651X~96!00506-5#

Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble

Abstract: A method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction. Systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail. Mutual nonlinear prediction also provides information on the directionality of the coupling between systems. Furthermore, the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony. Numerical examples studied include three classes of unidirectionally coupled systems: systems with identical parameters, nonidentical parameters, and stochastic driving of a nonlinear system. This technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool. The interrelationships examined include single neuron unit firing, the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex, and intracellular measurements of integrated excitatory postsynaptic potentials ~EPSP’s !. Dynamical interdependence, perhaps generalized synchrony, was identified in this neuronal network between simultaneous single unit firings, between units and the population, and between units and intracellular EPSP’s.

Experimental investigation of the melting transition of the plasma crystal.

Abstract: Measurements of the melting transition of a Coulomb crystal consisting of dust particles immersed in an rf parallel plate discharge in helium were performed. The dust crystal is shown to be solid at higher gas pressure (120 Pa) and low discharge power (10--20 W). Reducing the gas pressure or increasing the discharge power leads to fluid states of the dust ensemble. Even gaslike states are observed at low pressures of about 40 Pa. The transition is attributed to an increasing effective particle temperature. The phase transition is compared with one-component-plasma and Yukawa models, and with basic predictions of theories for two-dimensional melting. \textcopyright{} 1996 The American Physical Society.

Method for generating long-range correlations for large systems.

Abstract: We propose a new method to generate a sequence of random numbers with long-range power-law correlations that overcomes known difficulties associated with large systems. The new method presents an improvement on the commonly-used methods. We apply the algorithm to generate enhanced diffusion, isotropic and anisotropic self-affine surfaces, and isotropic and anisotropic correlated percolation.

Model for force fluctuations in bead packs.

Abstract: We study theoretically the complex network of forces that is responsible for the static structure and properties of granular materials. We present detailed calculations for a model in which the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead of the pile. We compare our results for the force distribution function for this model, including exact results for certain contact angle probability distributions, with numerical simulations of force distributions in random sphere packings. This model reproduces many aspects of the force distribution observed both in experiment and in numerical simulations of sphere packings. Our model is closely related to some that have been studied in the context of self-organized criticality. We present evidence that in the force distribution context, "critical" power-law force distributions occur only when a parameter (hidden in other interpretations) is tuned. Our numerical, mean field, and exact results all indicate that for almost all contact distributions the distribution of forces decays exponentially at large forces.

Tortuous flow in porous media

Abstract: The concept of tortuosity of fluid flow in porous media is discussed. A lattice-gas cellular automaton method is applied to solve the flow of a Newtonian uncompressible fluid in a two-dimensional porous substance constructed by randomly placed rectangles of equal size and with unrestricted overlap. A clear correlation between the average tortuosity of the flow paths and the porosity of the substance has been found. \textcopyright{} 1996 The American Physical Society.

Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices

Abstract: We study statistical properties of the ensemble of large N\ifmmode\times\else\texttimes\fi{}N random matrices whose entries ${\mathit{H}}_{\mathit{ij}}$ decrease in a power-law fashion ${\mathit{H}}_{\mathit{ij}}$\ensuremath{\sim}|i-j${\mathrm{|}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$. Mapping the problem onto a nonlinear \ensuremath{\sigma} model with nonlocal interaction, we find a transition from localized to extended states at \ensuremath{\alpha}=1. At this critical value of \ensuremath{\alpha} the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson statistics. These features are reminiscent of those typical of the mobility edge of disordered conductors. We find a continuous set of critical theories at \ensuremath{\alpha}=1, parametrized by the value of the coupling constant of the \ensuremath{\sigma} model. At \ensuremath{\alpha}g1 all states are expected to be localized with integrable power-law tails. At the same time, for 13/2 the wave packet spreading at a short time scale is superdiffusive: 〈|r|〉\ensuremath{\sim}${\mathit{t}}^{1/(2\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{-}}1)}$, which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/21 the statistical properties of eigenstates are similar to those in a metallic sample in d=(\ensuremath{\alpha}-1/2${)}^{\mathrm{\ensuremath{-}}1}$ dimensions. Finally, the region \ensuremath{\alpha}1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (\ensuremath{\alpha}=0). The theoretical predictions are compared with results of numerical simulations. \textcopyright{} 1996 The American Physical Society.

Aperiodic stochastic resonance.

Abstract: Stochastic resonance (SR) is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise. Recently, we presented a method and theory for characterizing SR-type behavior in excitable systems with aperiodic (i.e., broadband) input signals [Phys. Rev. E 52, R3321(1995)]. We coined the term aperiodic stochastic resonance (ASR) to describe this general type of behavior. In that earlier study, we demonstrated ASR in the FitzHugh-Nagumo neuronal model. Here we demonstrate ASR in three additional systems: a bistable-well system, an integrate-and-fire neuronal model, and the Hodgkin-Huxley (HH) neuronal model. We present computational and theoretical results for each system. In the context of the HH model, we develop a general theory for ASR in excitable membranes. This work clearly shows that SR-type behavior is not limited to systems with periodic inputs. Thus, in general, noise can serve to enhance the response of a nonlinear system to a weak input signal, regardless of whether the signal is periodic or aperiodic. \textcopyright{} 1996 The American Physical Society.

Shape transformations of vesicles with intramembrane domains

Abstract: Phase separation within the lipid bilayer of vesicles can lead to the formation of domains that affect the equilibrium shape of these vesicles. As a result of the competition between the bending energy of the bilayer and the line energy of the domain boundaries, the domains induce the formation of buds if their size is sufficiently large. This phenomenon of domain-induced budding is studied both for freely adapting and for fixed volume. The phase diagrams show that the constraint on the volume acts against the budding process but will in general not suppress it. In many situations, domain-induced budding leads to limit shapes for which the bud consists of a closed sphere that is connected to the ``mother'' vesicle by an infinitesimal neck. This neck is characterized by a general and simple neck condition for the mean curvature of the membrane segments adjacent to the neck. Budding also occurs if the Gaussian bending energy is taken into account. The effect of the Gaussian curvature energy is to change the structure and the stability of those shapes that exhibit small necks. \textcopyright{} 1996 The American Physical Society.

Thermodynamic description of the relaxation of two-dimensional turbulence using Tsallis statistics

Abstract: Two-dimensional Euler turbulence and drift turbulence in a pure-electron plasma column have been experimentally observed to relax to metaequilibrium states that do not maximize the Boltzmann entropy, but rather seem to minimize enstrophy. We show that a recent generalization of thermodynamics and statistics due to Tsallis [Phys. Lett. A 195, 329 (1994); J. Stat. Phys. 52, 479 (1988)] is capable of explaining this phenomenon in a natural way. In particular, the maximization of the generalized entropy ${S}_{q}$ with $q=\frac{1}{2}$ for the pure-electron plasma column leads to precisely the same profiles predicted by the restricted minimum enstrophy theory of Huang and Driscoll [Phys. Rev. Lett. 72, 2187 (1994)]. These observations make possible the construction of a comprehensive thermodynamic description of two-dimensional turbulence.

Partial synchronization in populations of pulse-coupled oscillators.

Abstract: I study the long-term behavior of populations of nonlinear oscillators with all-to-all, noninstantaneous, pulse coupling. With fast enough excitatory coupling both the fully synchronized and the asynchronous state are unstable. In this case individual units fire quasiperiodically even though the network as a whole shows a periodic firing pattern. The behavior of networks with three or more units is different in this regard from that of two-unit networks. With inhibitory coupling the network can break up into a variable number of fully synchronized clusters. For fast inhibition the number of clusters tends to be large, while the number of clusters is smaller for slow inhibition. \textcopyright{} 1996 The American Physical Society.

Stochastic foundations of fractional dynamics

Abstract: It is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance (fractal walks), thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk [R. Hilfer and L. Anton, Phys. Rev. E 51, R848 (1995)].

Supercooled water and the kinetic glass transition.

Abstract: We present a molecular-dynamics study of the self-dynamics of water molecules in deeply supercooled liquid states. We find that the decay of single-particle dynamics correlation functions is characterized by a fast initial relaxation toward a plateau and by a region of self-similar dynamics, followed at late times by a stretched exponential decay. We interpret such results in the framework of the mode-coupling theory for supercooled liquids. We relate the apparent anomalies of the transport coefficients in water on lowering the temperature to the formation of cages and to the associated slow dynamics resulting from the presence of long-lived molecular cages. The so-called critical Angell temperature in supercooled water could thus be interpreted as kinetic glass transition temperature, relaxing the need of a thermodynamic singularity for the explanation of the anomalies of liquid water.