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PubMed2006

Fractional translational diffusion of a Brownian particle in a double well potential.

The fractional translational diffusion of a particle in a double-well potential (excluding inertial effects) is considered. The position correlation function and its spectrum are evaluated using a fractional probability density diffusion equation (based on the diffusion limit of a fractal time random walk). Exact and approximate solutions for the dynamic susceptibility describing the position response to a small external field are obtained. The exact solution is given by matrix continued fractions while the approximate solution relies on the exponential separation of the time scales of the fast "intrawell" and low overbarrier relaxation processes associated with the bistable potential. It is shown that knowledge of the characteristic relaxation times for normal diffusion allows one to predict accurately the anomalous relaxation behavior of the system for all relevant time scales.

Journal Article↗
PubMed2006

Phase transitions in systems possessing shock solutions.

Recently it has been shown that there are three families of stochastic one-dimensional nonequilibrium lattice models for which the single-shock measures form an invariant subspace of the states of these models. Here, both the stationary states and dynamics of single-shocks on a one-dimensional lattice are studied. This is done for both an infinite lattice and a finite lattice with boundaries. It is seen that these models possess both static and dynamical phase transitions. The static phase transition is the well-known low-high density phase transition for the asymmetric simple exclusion process. The branching-coalescing random walk and asymmetric Kawasaki-Glauber process models also show the same phase transition. Double-shocks on a one-dimensional lattice are also investigated. It is shown that at the stationary state the contribution of double-shocks with higher width becomes small, and the main contribution comes from thin double-shocks.

Journal Article↗
PubMed2006

Velocity and diffusion coefficient of A + A <--> A reaction fronts in one dimension.

We study front propagation in the reversible reaction-diffusion system A + A <--> A on a one-dimensional lattice. Extending the idea of leading particle in studying the motion of the front we write a master equation in the stochastically moving frame attached to this particle. This approach provides a systematic way to improve on estimates of front speed obtained earlier. We also find that the leading particle performs a correlated random walk and this correlation needs to be taken into account to get the correct the value of the front diffusion coefficient.

Journal Article↗
PubMed2006

Condition for intracellular adaptive dynamics for chemotaxis.

Chemotaxis is ubiquitously performed by bacteria. They sense and move toward a region with a higher concentration of an attractive chemical by changing the rate of tumbling for random walk. We numerically studied several models with internal adaptive dynamics to examine the validity of the condition for chemotaxis proposed by Oosawa and Nakaoka [J. Theor. Biol. 66, 747 (1977)], which states that the time scale of tumbling frequency must be smaller than that of adaptation and greater than that of sensing. Suitable renormalization of the time scales showed that the condition holds for a variety of environments and for both short- and long-term behavior.

Adaptation, Physiological↗
PubMed2006

Classical momentum diffusion in double-delta-kicked particles.

We investigate the classical chaotic diffusion of atoms subjected to pairs of closely spaced pulses ("kicks") from standing waves of light (the 2delta-KP ). Recent experimental studies with cold atoms implied an underlying classical diffusion of a type very different from the well-known paradigm of Hamiltonian chaos, the standard map. The kicks in each pair are separated by a small time interval E<<1, which together with the kick strength K, characterizes the transport. Phase space for the 2delta-KP is partitioned into momentum "cells" partially separated by momentum-trapping regions where diffusion is slow. We present here an analytical derivation of the classical diffusion for a 2delta-KP including all important correlations which were used to analyze the experimental data. We find an asymptotic (t-->infinity) regime of "hindered" diffusion: while for the standard map the diffusion rate, for K>>1 , D approximately K(2)/2[1-2J(2)(K)...] oscillates about the uncorrelated rate D(0)=K(2)/2, we find analytically, that the 2delta-KP can equal, but never diffuses faster than, a random walk rate. We argue this is due to the destruction of the important classical "accelerator modes" of the standard map. We analyze the experimental regime 0.1 less or approximately KE less or approximately 1 , where quantum localization lengths L approximately Planck's (-0.75) are affected by fractal cell boundaries. We find an approximate asymptotic diffusion rate D proportional to K(3)E, in correspondence to a D proportional to K(3) regime in the standard map associated with the "golden-ratio" cantori.

Journal Article↗
PubMed2006

Dynamical network model of infective mobile agents.

A dynamical network (consisting of a time-evolving wiring of interactions among a group of random walkers) is introduced to model the spread of an infectious disease in a population of mobile individuals. We investigate the main properties of this model, and show that peculiar features arise when individuals are allowed to perform long-distance jumps. Such peculiarities are captured and conveniently quantified by a series of appropriate parameters able to highlight the structural differences emerging in the networks when long-distance jumps are combined with local random walk processes.

Journal Article↗
PubMed2006

Exact solution of a generalized Kramers-Fokker-Planck equation retaining retardation effects.

In order to describe non-Gaussian kinetics in weakly damped systems, the concept of continuous time random walks is extended to particles with finite inertia. One thus obtains a generalized Kramers-Fokker-Planck equation, which retains retardation effects, i.e., nonlocal couplings in time and space. It is shown that despite this complexity, exact solutions of this equation can be given in terms of superpositions of Gaussian distributions with varying variances. In particular, the long-time behavior of the respective low-order moments is calculated.

Journal Article↗
PubMed2000

Does hard core interaction change absorbing-type critical phenomena?

It has been generally believed that hard core interaction is irrelevant to absorbing-type critical phenomena because the particle density is so low near an absorbing phase transition. We study the effect of hard core interaction on the N-species branching annihilating random walks with two offspring and report that hard core interaction drastically changes the absorbing-type critical phenomena in a nontrivial way. Through a Langevin equation-type approach, we predict analytically the values of the scaling exponents, nu( perpendicular) = 2, z = 2, alpha = 1/2, and beta = 2 in one dimension for all N>1. Direct numerical simulations confirm our prediction. When the diffusion coefficients for different species are not identical, nu( perpendicular) and beta vary continuously with the ratios between the coefficients.

Journal Article↗
PubMed2001

Spatial persistence of fluctuating interfaces.

We show that the probability, P0(l), that the height of a fluctuating (d+1)-dimensional interface in its steady state stays above its initial value up to a distance l, along any linear cut in the d-dimensional space, decays as P0(l) approximately l(theta). Here straight theta is a "spatial" persistence exponent, and takes different values, straight theta(s) or straight theta(0), depending on how the point from which l is measured is specified. These exponents are shown to map onto corresponding temporal persistence exponents for a generalized d = 1 random-walk equation. The exponent straight theta(0) is nontrivial even for Gaussian interfaces.

Journal Article↗
PubMed2001

Best possible strategy for finding ground states.

Finding the ground state of a system with a complex energy landscape is important for many physical problems including protein folding, spin glasses, chemical clusters, and neural networks. Such problems are usually solved by heuristic search methods whose efficacy is judged by empirical performance on selected examples. We present a proof that, within the large class of algorithms that simulate a random walk on the landscape, threshold accepting is the best possible strategy. In particular, it can perform better than simulated annealing and Tsallis statistics. Our proof is the first example of a provably optimal strategy in this area.

Algorithms↗
PubMed2001

Spatiotemporal scaling of solar surface flows.

The sun provides an excellent natural laboratory for nonlinear phenomena. We use motions of magnetic bright points on the solar surface, at the smallest scales yet observed, to study the small scale dynamics of the photospheric plasma. The paths of the bright points are analyzed within a continuous time random walk framework. Their spatial and temporal scaling suggests that the observed motions are the walks of imperfectly correlated tracers on a turbulent fluid flow in the lanes between granular convection cells.

Journal Article↗
PubMed2001

Time-Inhomogeneous Fokker-Planck Equation for Wave Distributions in the Abelian Sandpile Model.

The time and size distributions of the waves of topplings in the Abelian sandpile model are expressed as the first arrival at the origin distribution for a scale invariant, time-inhomogeneous Fokker-Plank equation. Assuming a linear conjecture for the time inhomogeneity exponent as a function of a loop-erased random walk (LERW) critical exponent, suggested by numerical results, this approach allows one to estimate the lower critical dimension of the model and the exact value of the critical exponent for LERW in three dimensions. The avalanche size distribution in two dimensions is found to be the difference between two closed power laws.

Journal Article↗
PubMed2001

Front propagation into an unstable state of reaction-transport systems.

We studied the propagation of traveling fronts into an unstable state of the reaction-transport systems involving integral transport. By using a hyperbolic scaling procedure and singular perturbation techniques, we determined a Hamiltonian structure of reaction-transport equations. This structure allowed us to derive asymptotic formulas for the propagation rate of a reaction front. We showed that the macroscopic dynamics of the front are "nonuniversal" and depend on the choice of the underlying random walk model for the microscopic transport process.

Journal Article↗
PubMed2001

Topological description of the aging dynamics in simple glasses.

We numerically investigate the aging dynamics of a monatomic Lennard-Jones glass, focusing on the topology of the potential energy landscape which, to this aim, has been partitioned in basins of attraction of stationary points (saddles and minima). The analysis of the stationary points visited during the aging dynamics shows the existence of two distinct regimes: (i) at short times the system visits basins of saddles whose energies and orders decrease with t; (ii) at long times the system mainly lies in basins pertaining to minima of slowly decreasing energy. The long time dynamics can be represented by a simple random walk on a network of minima with a jump probability proportional to the inverse of the waiting time.

Journal Article↗
PubMed2001

Thermodynamic formalism of the harmonic measure of diffusion limited aggregates: phase transition.

We study the nature of the phase transition in the multifractal formalism of the harmonic measure of diffusion limited aggregates. Contrary to previous work that relied on random walk simulations or ad hoc models to estimate the low probability events of deep fjord penetration, we employ the method of iterated conformal maps to obtain an accurate computation of the probability of the rarest events. We resolve probabilities as small as 10(-35). We show that the generalized dimensions D(q) are infinite for q<q*, where q* = -0.18+/-0.04. In the language of f(alpha) this means that alpha(max) is finite. We present a converged f(alpha) curve.

Journal Article↗
PubMed2001

Diffusion with random distribution of static traps.

The survival probability P(c,t) of a random walk of t steps with static traps at concentration c is studied in two and three dimensions by an efficient Monte Carlo method based on a mapping onto a polymer model. On the basis of the theoretical work of Donsker and Varadhan [Commun. Pure Appl. Math. 28, 525 (1975); 32, 721 (1979)] and of Rosenstock [J. Math. Phys. (N.Y.) 11, 487 (1970)] one expects a data collapse for -ln[P(c,t)]/ln(t) plotted vs square root of [lambda t]/ln(t) [with lambda = -ln(1-c)], in two dimensions, and for -t(-1/3)ln[P(c,t)] vs t(2/3)lambda in three dimensions. These predictions are well supported by the Monte Carlo results.

Journal Article↗
PubMed2002

Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies.

An important application involving two-species reaction-diffusion systems relates to the problem of finding the best statistical strategy for optimizing the encounter rate between organisms. We investigate the general problem of how the encounter rate depends on whether organisms move in Lévy or Brownian random walks. By simulating a limiting generalized searcher-target model (e.g., predator-prey, mating partner, pollinator-flower), we find that Lévy walks confer a significant advantage for increasing encounter rates when the searcher is larger or moves rapidly relative to the target, and when the target density is low.

Algorithms↗
PubMed2002

Universal attractors of reversible aggregate-reorganization processes.

We analyze a general class of reversible aggregate-reorganization processes. These processes are shown to exhibit globally attracting equilibrium distributions, which are universal, i.e., identical for large classes of models. Furthermore, the analysis implies that, for studies of equilibrium properties of any such process, computationally expensive reorganization dynamics such as random walks can be replaced by more efficient yet simpler methods. As a particular application, our results explain the recent observation of the formation of similar fractal aggregates from different initial structures by diffusive reorganization [M. Filoche and B. Sapoval, Phys. Rev. Lett. 85, 5118 (2000)].

Cell Aggregation↗