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At least 1,153 records · Page 64Linked to original sources
PubMed2005

Anisotropic diffusion limited aggregation in three dimensions: universality and nonuniversality.

We explore the macroscopic consequences of lattice anisotropy for diffusion limited aggregation (DLA) in three dimensions. Simple cubic and bcc lattice growths are shown to approach universal asymptotic states in a coherent fashion, and the approach is accelerated by the use of noise reduction. These states are strikingly anisotropic dendrites with a rich hierarchy of structure. For growth on an fcc lattice, our data suggest at least two stable fixed points of anisotropy, one matching the bcc case. Hexagonal growths, favoring six planar and two polar directions, appear to approach a line of asymptotic states with continuously tunable polar anisotropy. The more planar of these growths visually resembles real snowflake morphologies. Our simulations use a new and dimension-independent implementation of the DLA model. The algorithm maintains a hierarchy of sphere coverings of the growth, supporting efficient random walks onto the growth by spherical moves. Anisotropy was introduced by restricting growth to certain preferred directions.

Journal Article↗
PubMed2005

Dynamic self-consistent field theory for unentangled homopolymer fluids.

We present a lattice formulation of a dynamic self-consistent field (DSCF) theory that is capable of resolving interfacial structure, dynamics, and rheology in inhomogeneous, compressible melts and blends of unentangled homopolymer chains. The joint probability distribution of all the Kuhn segments in the fluid, interacting with adjacent segments and walls, is approximated by a product of one-body probabilities for free segments interacting solely with an external potential field that is determined self-consistently. The effect of flow on ideal chain conformations is modeled with finitely extensible, nonlinearly elastic dumbbells in the Peterlin approximation, and related to stepping probabilities in a random walk. Free segment and stepping probabilities generate statistical weights for chain conformations in a self-consistent field, and determine local volume fractions of chain segments. Flux balance across unit lattice cells yields mean field transport equations for the evolution of free segment probabilities and of momentum densities on the Kuhn length scale. Diffusive and viscous contributions to the fluxes arise from segmental hops modeled as a Markov process, with transition rates reflecting changes in segmental interaction, kinetic energy, and entropic contributions to the free energy under flow. We apply the DSCF equations to study both transient and steady-state interfacial structure, flow, and rheology in a sheared planar channel containing either a one-component melt or a phase-separated, two-component blend.

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PubMed2005

Morphogen gradient formation in a complex environment: an anomalous diffusion model.

Current models of morphogen-induced patterning assume that morphogens undergo normal, or Fickian, diffusion, although the validity of this assumption has never been examined. Here we argue that the interaction of morphogens with the complex extracellular surrounding may lead to anomalous diffusion. We present a phenomenological model that captures this interaction, and derive the properties of the morphogen profile under conditions of anomalous (non-Fickian) diffusion. In this context we consider the continuous time random walk formalism and extend its application to account for degradation of morphogen particles. We show that within the anomalous diffusion model, morphogen profiles are fundamentally distinct from the corresponding Fickian profiles. Differences were found in several key aspects, including the role of degradation in determining the profile, the rate by which it spreads in time and its long-term behavior. We analyze the effect of an abrupt change in the extracellular environment on the concentration profiles. Furthermore, we discuss the robustness of the morphogen distribution to fluctuations in morphogen production rate, and describe a feedback mechanism that can buffer such fluctuations. Our study also provides rigorous criteria to distinguish experimentally between Fickian and anomalous modes of morphogen transport.

Algorithms↗
PubMed2005

Scaling and data collapse for the mean exit time of asset prices.

We study theoretical and empirical aspects of the mean exit time (MET) of financial time series. The theoretical modeling is done within the framework of continuous time random walk. We empirically verify that the mean exit time follows a quadratic scaling law and it has associated a prefactor which is specific to the analyzed stock. We perform a series of statistical tests to determine which kind of correlation are responsible for this specificity. The main contribution is associated with the autocorrelation property of stock returns. We introduce and solve analytically both two-state and three-state Markov chain models. The analytical results obtained with the two-state Markov chain model allows us to obtain a data collapse of the 20 measured MET profiles in a single master curve.

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PubMed2005

Reliability of the 0-1 test for chaos.

In time series analysis, it has been considered of key importance to determine whether a complex time series measured from the system is regular, deterministically chaotic, or random. Recently, Gottwald and Melbourne have proposed an interesting test for chaos in deterministic systems. Their analyses suggest that the test may be universally applicable to any deterministic dynamical system. In order to fruitfully apply their test to complex experimental data, it is important to understand the mechanism for the test to work, and how it behaves when it is employed to analyze various types of data, including those not from clean deterministic systems. We find that the essence of their test can be described as to first constructing a random walklike process from the data, then examining how the variance of the random walk scales with time. By applying the test to three sets of data, corresponding to (i) 1/falpha noise with long-range correlations, (ii) edge of chaos, and (iii) weak chaos, we show that the test mis-classifies (i) both deterministic and weakly stochastic edge of chaos and weak chaos as regular motions, and (ii) strongly stochastic edge of chaos and weak chaos, as well as 1/falpha noise as deterministic chaos. Our results suggest that, while the test may be effective to discriminate regular motion from fully developed deterministic chaos, it is not useful for exploratory purposes, especially for the analysis of experimental data with little a priori knowledge. A few speculative comments on the future of multiscale nonlinear time series analysis are made.

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PubMed2005

Patterns and localized structures in population dynamics.

Patterns, fronts, and localized structures of a prototypical model for population dynamics interaction are studied. The physical content of the model is the coexistence of a simple random walk for the motion of the individuals with a nonlinearity in the competitive struggle for resources which simultaneously stresses the Allee effect and interaction at a distance. Mathematically, the model is variational and exhibits coexistence between different stable extended states. Solutions are obtained, the phase diagram is constructed, and the emergence of localized structures is investigated.

Biophysics↗
PubMed2005

Nuclear-magnetic-resonance diffusion simulations in two phases in porous media.

Time-dependent diffusion simulations which can be measured by nuclear magnetic resonance (NMR) were numerically performed in consolidated reconstructed porous media saturated by two immobile fluids. The phase distributions were obtained by an immiscible lattice Boltzmann technique which incorporates interfacial tension and wetting. The apparent diffusion coefficient in each fluid was determined by a random walk algorithm. Permeability and conductivity tensors were calculated by finite-difference schemes. The major properties valid for a single phase could be generalized to two phases. First, the characteristic length Lambda introduced by Johnson et al.[Phys. Rev. Lett. 57, 2564 (1986)] is of the order of twice the phase volume to surface ratio. Second, the apparent diffusion coefficients for all porosities, saturations, and phases can be represented by a single dimensionless curve.

Journal Article↗
PubMed2005

Extremal dynamics on complex networks: analytic solutions.

The Bak-Sneppen model displaying punctuated equilibria in biological evolution is studied on random complex networks. By using the rate equation and the random walk approaches, we obtain the analytic solution of the fitness threshold xc to be 1/((k)f+1), where (k)f=(k2)/(k) (=(k)) in the quenched (annealed) updating case, where kn is the nth moment of the degree distribution. Thus, the threshold is zero (finite) for the degree exponent gamma<3 (gamma>3) for the quenched case in the thermodynamic limit. The theoretical value xc fits well to the numerical simulation data in the annealed case only. Avalanche size, defined as the duration of successive mutations below the threshold, exhibits a critical behavior as its distribution follows a power law, Pa(s) approximately s(-3/2).

Journal Article↗
PubMed2005

Uncovering collective listening habits and music genres in bipartite networks.

In this paper, we analyze web-downloaded data on people sharing their music library, that we use as their individual musical signatures. The system is represented by a bipartite network, nodes being the music groups and the listeners. Music groups' audience size behaves like a power law, but the individual music library size is an exponential with deviations at small values. In order to extract structures from the network, we focus on correlation matrices, that we filter by removing the least correlated links. This percolation idea-based method reveals the emergence of social communities and music genres, that are visualized by a branching representation. Evidence of collective listening habits that do not fit the neat usual genres defined by the music industry indicates an alternative way of classifying listeners and music groups. The structure of the network is also studied by a more refined method, based upon a random walk exploration of its properties. Finally, a personal identification-community imitation model for growing bipartite networks is outlined, following Potts ingredients. Simulation results do reproduce quite well the empirical data.

Journal Article↗
PubMed2006

Brownian particle having a fluctuating mass.

We focus on the dynamics of a Brownian particle whose mass fluctuates. First we show that the behavior is similar to that of a Brownian particle moving in a fluctuating medium, as studied by Beck [Phys. Rev. Lett. 87, 180601 (2001)]. By performing numerical simulations of the Langevin equation, we check the theoretical predictions derived in the adiabatic limit, i.e. when the mass fluctuation time scale is much larger than the time for reaching the local equilibrium, and study deviations outside this limit. We compare the mass velocity distribution with truncated Tsallis distributions [J. Stat. Phys. 52, 479 (1988)] and find excellent agreement if the masses are chi-squared distributed. We also consider the diffusion of the Brownian particle by studying a Bernoulli random walk with fluctuating walk length in one dimension. We observe the time dependence of the position distribution kurtosis and find interesting behaviors. We point out a few physical cases, where the mass fluctuation problem could be encountered as a first approximation for agglomeration-fracture nonequilibrium processes.

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PubMed2006

Simplest piston problem. I. Elastic collisions.

We study the dynamics of three elastic particles in a finite interval where two light particles are separated by a heavy "piston." The piston undergoes surprisingly complex motion that is oscillatory at short time scales but seemingly chaotic at longer scales. The piston also makes long-duration excursions close to the ends of the interval that stem from the breakdown of energy equipartition. Many of these dynamical features can be understood by mapping the motion of three particles on the line onto the trajectory of an elastic billiard in a highly skewed tetrahedral region. We exploit this picture to construct a qualitative random walk argument that predicts a power-law tail, with exponent -3/2, for the distribution of time intervals between successive piston crossings of the interval midpoint. These predictions are verified by numerical simulations.

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PubMed2006

Current and universal scaling in anomalous transport.

Anomalous transport in tilted periodic potentials is investigated within the framework of the fractional Fokker-Planck dynamics and the underlying continuous time random walk. The analytical solution for the stationary, anomalous current is obtained in closed form. We derive a universal scaling law for anomalous diffusion occurring in tilted periodic potentials. This scaling relation is corroborated with precise numerical studies covering wide parameter regimes and different shapes for the periodic potential, being either symmetric or ratchetlike.

Journal Article↗
PubMed2006

Space-fractional advection-diffusion and reflective boundary condition.

Anomalous diffusive transport arises in a large diversity of disordered media. Stochastic formulations in terms of continuous time random walks (CTRWs) with transition probability densities showing space- and/or time-diverging moments were developed to account for anomalous behaviors. A broad class of CTRWs was shown to correspond, on the macroscopic scale, to advection-diffusion equations involving derivatives of noninteger order. In particular, CTRWs with Lévy distribution of jumps and finite mean waiting time lead to a space-fractional equation that accounts for superdiffusion and involves a nonlocal integral-differential operator. Within this framework, we analyze the evolution of particles performing symmetric Lévy flights with respect to a fluid moving at uniform speed . The particles are restricted to a semi-infinite domain limited by a reflective barrier. We show that the introduction of the boundary condition induces a modification in the kernel of the nonlocal operator. Thus, the macroscopic space-fractional advection-diffusion equation obtained is different from that in an infinite medium.

Journal Article↗
PubMed2006

Occurrence of normal and anomalous diffusion in polygonal billiard channels.

From extensive numerical simulations, we find that periodic polygonal billiard channels with angles which are irrational multiples of pi generically exhibit normal diffusion (linear growth of the mean squared displacement) when they have a finite horizon, i.e., when no particle can travel arbitrarily far without colliding. For the infinite horizon case we present numerical tests showing that the mean squared displacement instead grows asymptotically as t ln t. When the unit cell contains accessible parallel scatterers, however, we always find anomalous super-diffusion, i.e., power-law growth with an exponent larger than . This behavior cannot be accounted for quantitatively by a simple continuous-time random walk model. Instead, we argue that anomalous diffusion correlates with the existence of families of propagating periodic orbits. Finally we show that when a configuration with parallel scatterers is approached there is a crossover from normal to anomalous diffusion, with the diffusion coefficient exhibiting a power-law divergence.

Journal Article↗
PubMed2006

Kardar-Parisi-Zhang equation in the weak noise limit: pattern formation and upper critical dimension.

We extend the previously developed nonperturbative weak noise scheme, applied to the noisy Burgers equation in one dimension, to the Kardar-Parisi-Zhang equation for a growing interface in arbitrary dimensions. By means of the Cole-Hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes. In the Cole-Hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear Schrödinger equation, subsequently boosted to finite velocity by a Galilei transformation. We discuss the dynamics of the pattern formation and, briefly, the superimposed linear modes. Implementing the stochastic interpretation we discuss kinetic transitions and in particular the preliminary scaling properties pertaining to the pair mode or dipole sector. In the dipole sector we obtain the Hurst exponent H=(3-d)/(4-d) or dynamic exponent Zdip(4-d)/(3-d) for the random walk of growth modes. Below d=3 the dipole growth modes show anomalous diffusion, above d=3 the dipole growth modes freeze. Finally, applying Derrick's theorem based on constrained minimization we show that the upper critical dimension is d=4 in the sense that growth modes cease to exist above this dimension.

Journal Article↗
PubMed2006

Genetic attack on neural cryptography.

Different scaling properties for the complexity of bidirectional synchronization and unidirectional learning are essential for the security of neural cryptography. Incrementing the synaptic depth of the networks increases the synchronization time only polynomially, but the success of the geometric attack is reduced exponentially and it clearly fails in the limit of infinite synaptic depth. This method is improved by adding a genetic algorithm, which selects the fittest neural networks. The probability of a successful genetic attack is calculated for different model parameters using numerical simulations. The results show that scaling laws observed in the case of other attacks hold for the improved algorithm, too. The number of networks needed for an effective attack grows exponentially with increasing synaptic depth. In addition, finite-size effects caused by Hebbian and anti-Hebbian learning are analyzed. These learning rules converge to the random walk rule if the synaptic depth is small compared to the square root of the system size.

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PubMed2006

Fractional Fokker-Planck dynamics: Numerical algorithm and simulations.

Anomalous transport in a tilted periodic potential is investigated numerically within the framework of the fractional Fokker-Planck dynamics via the underlying continuous-time random walk. An efficient numerical algorithm is developed which is applicable for an arbitrary potential. This algorithm is then applied to investigate the fractional current and the corresponding nonlinear mobility in different washboard potentials. Normal and fractional diffusion are compared through their time evolution of the probability density in state space. Moreover, we discuss the stationary probability density of the fractional current values.

Journal Article↗
PubMed2006

Toy model of fractional transport of cancer cells due to self-entrapping.

A simple mathematical model is proposed to study the influence of cell fission on transport. The model describes fractional, in time, tumor development, which is a one-dimensional continuous-time random walk. The model is relevant for consideration of both solid and diffusive cancers.

Cell Movement↗