Advection and dispersion in time and space

Previous work showed how moving particles that rest along their trajectory lead to time-nonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions to the equation are obtained by subordination. The form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly.     

Random walks with infinite spatial and temporal moments

The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial, and temporal moments of the memory are infinite. The asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated. The resulting behavior includes localized, diffusive, wavelike, and Levy's stable laws for the appropriate scaled variable. We show that an infinite mean waiting time can lead to long time diffusive behavior, while a finite mean waiting time is not sufficient to ensure the same.     

Unified scaling law for earthquakes

We show that the distribution of waiting times between earthquakes occurring in California obeys a simple unified scaling law valid from tens of seconds to tens of years. The short time clustering, commonly referred to as aftershocks, is nothing but the short time limit of the general hierarchical properties of earthquakes. There is no unique operational way of distinguishing between main shocks and aftershocks. In the unified law, the Gutenberg-Richter b value, the exponent -1 of the Omori law for aftershocks, and the fractal dimension d(f) of earthquakes appear as critical indices.     

Transients and asymptotics in granular phase space

The dynamics of dissipative dynamical systems can be described by the sequential appearance of two different regimes. From a given initial condition, one first observes transient behavior characterized by a high degree of contraction of volumes in phase space. This is followed by an asymptotic regime with one or several attractors into which trajectories inject after long times. There is however, no sharp crossover between these two regimes and the identification of either one depends on the precision of measurement. In order to investigate these issues, we studied the dynamics of contracting integer maps. We found out that for the cases which in the continuum limit correspond to bifurcations, transients consists of two regimes sharply separated by a crossover point which displays universal scaling with the size of the set. Moreover, their average lengths display power law dependence on the accuracy of their measurement. This behavior persists away from bifurcation but with a different scaling law. In addition, we studied deterministic diffusion on finite sets and obtained analytic expressions for the mean square displacement in the long time limit.     

Scaling properties of geometric parallelization

We present a universal scaling law for all geometrically parallelized computer simulation algorithms. For algorithms with local interaction laws we calculate the scaling exponents for zero and infinite lattice size. The scaling is tested on local (cellular automata, Metropolis Ising) as well as cluster (Swendsen-Wang) algorithms. The practical aspects of the scaling properties lead to a simple recipe for finding the optimum number of processors to be used for the parallel simulation of a particular system.     

From diffusion to anomalous diffusion: a century after Einstein's Brownian motion

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.     

Violation of the entropic area law for fermions

We investigate the scaling of the entanglement entropy in an infinite translational invariant fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. Thus, in contrast with analogous bosonic systems, the entropic area law is violated for fermions. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven that the presented scaling law holds whenever the Fermi surface is finite. This is, in particular, true for all ground states of Hamiltonians with finite range interactions.     

Correlated dynamics in human printing behavior

Arrival times of requests to print in a student laboratory were analyzed. Inter-arrival times between subsequent requests follow a universal scaling law relating time intervals and the size of the request, indicating a scale invariant dynamics with respect to the size. The cumulative distribution of file sizes is well-described by a modified power-law often seen in non-equilibrium critical systems. For each user, waiting times between their individual requests show long range dependence and are broadly distributed from seconds to weeks. All results are incompatible with Poisson models, and may provide evidence of critical dynamics associated with voluntary thought processes in the brain.     

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.     

Non-equilibrium dynamics of simple spherical spin models

We investigate the non-equilibrium dynamics of spherical spin models with two-spin interactions. For the exactly solvable models of the d-dimensional spherical ferromagnet and the spherical Sherrington-Kirkpatrick (SK) model the asymptotic dynamics has for large times and large waiting times the same formal structure. In the limit of large waiting times we find in both models an intermediate time scale, scaling as a power of the waiting time with an exponent smaller than one, and thus separating the time-translation-invariant short-time dynamics from the aging regime. It is this time scale on which the fluctuation-dissipation theorem is violated. Aging in these models is similar to that observed in spin glasses at the level of correlation functions, but different at the level of response functions, and thus different at the level of experimentally accessible quantities like thermoremanent magnetization. Received 22 April 1999     

Scalings of mixed-mode regimes in a simple polynomial three-variable model of nonlinear dynamical systems

We describe scaling laws for a control parameter for various sequences of bifurcations of the LSn mixed-mode regimes consisting of single large amplitude maximum followed by n small amplitude peaks. These regimes are obtained in a normalized version of a simple three-variable polynomial model that contains only one nonlinear cubic term. The period adding bifurcations for LSn patterns scales as 1/n at low n and as 1/n2 at sufficiently large values of n. Similar scaling laws 1/k at low k and 1/k2 at sufficiently high values of k describe the period adding bifurcations for complex k(LSn)(LS(n + 1)) patterns. A finite number of basic LSn patterns and infinite sequences of complex k(LSn)(LS(n + 1)) patterns exist in the model. Each periodic pattern loses its stability by the period doubling bifurcations scaled by the Feigenbaum law. Also an infinite number of the broken Farey trees exists between complex periodic orbits. A family of 1D return maps constructed from appropriate Poincaré sections is a very fruitful tool in studies of the dynamical system. Analysis of this family of maps supports the scaling laws found using the numerical integration of the model.     

Universal order statistics of random walks

We study analytically the order statistics of a time series generated by the positions of a symmetric random walk of n steps with step lengths of finite variance σ(2). We show that the statistics of the gap d(k,n) = M(k,n)-M(k+1,n) between the kth and the (k+1)th maximum of the time series becomes stationary, i.e., independent of n as n → ∞ and exhibits a rich, universal behavior. The mean stationary gap exhibits a universal algebraic decay for large k, ~d(k,∞)-/σ 1/sqrt[2πk], independent of the details of the jump distribution. Moreover, the probability density (pdf) of the stationary gap exhibits scaling, Pr(d(k,∞) = δ) ~/= (sqrt[k]/σ)P(δsqrt[k]/σ), in the regime δ~ (d(k,∞)). The scaling function P(x) is universal and has an unexpected power law tail, P(x) ~ x(-4) for large x. For δ> (d(k,∞)) the scaling breaks down and the pdf gets cut off in a nonuniversal way. Consequently, the moments of the gap exhibit an unusual multiscaling behavior.     

Algebraic decay in self-similar Markov chains

A continuous-time Markov chain is used to model motion in the neighborhood of a critical invariant circle for a Hamiltonian map. States in the infinite chain represent successive rational approximants to the frequency of the invariant circle. For the case of a noble frequency, the chain is self-similar and the nonlinear integral equation for the first passage time distribution is solved exactly. The asymptotic distribution is a power law times a function periodic in the logarithm of the time. For parameters relevant to the critical noble circle, the decay proceeds ast ^–4.05.     

Nonrenewal statistics in the catalytic activity of enzyme molecules at mesoscopic concentrations

Recent fluorescence spectroscopy measurements of single-enzyme kinetics have shown that enzymatic turnovers form a renewal stochastic process in which the inverse of the mean waiting time between turnovers follows the Michaelis-Menten equation. We study enzyme kinetics at physiologically relevant mesoscopic concentrations using a master equation. From the exact solution of the master equation we find that the waiting times are neither independent nor identically distributed, implying that enzymatic turnovers form a nonrenewal stochastic process. The inverse of the mean waiting time shows strong departure from the Michaelis-Menten equation. The waiting times between consecutive turnovers are anticorrelated, where short intervals are more likely to be followed by long intervals and vice versa. Correlations persist beyond consecutive turnovers indicating that multiscale fluctuations govern enzyme kinetics.     

Non-LTE transfer with complete redistribution. Scaling laws for a slab

Scaling laws for conservative scattering in a finite slab are extracted from an asymptotic analysis of the integral equation for the source function. The solution is separated into an interior and a surface boundary layer part. The matching between the two parts provides a scaling law for the surface value of the source function. When expressed in terms of a mean number of scatterings, this scaling law is generalizable to non-conservative scattering and the empirical formula of Jones and Skumanich is recovered.     

Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation

Acoustic waves in tissues and weakly attenuative fluids often have an attenuation parameter, alpha(omega), satisfying alpha(omega)= alpha0omegay in which alpha0 is a constant, omega is the frequency, and y is between 1 and 2. This power law attenuation is not predicted by the classical thermoviscous wave equation and researchers have proposed different modified viscous wave equations in which the loss term is a convolution operator or a fractional spatial or temporal derivative. In this paper, acoustic waves undergoing power law attenuation are modeled by a modification to the thermoviscous wave equation in which the time derivative of the viscous term is replaced by a fractional time derivative. An explicit time domain, finite element formulation leads to a stable algorithm capable of simulating axisymmetric, broadband acoustic pulses propagating through attenuative and dispersive media. The algorithm does not depend on the Born approximation, long wavelength limit, or plane wave assumptions. The algorithm is validated for planar and focused transducers and results include radiation patterns from a viscous scatterer in a lossless background and signals reflected from a viscous layer. The program can be used to determine scattering parameters for large, strong, possibly viscous scatterers, in either a lossless or viscous background, for which analytic results are scarce.     

Continuous time random walk with jump length correlated with waiting time

A coupled continuous time random walk (CTRW) model is proposed, in which the jump length of a walker is correlated with waiting time. The power law distribution is chosen as the probability density function of waiting time and the Gaussian-like distribution as the probability density function of jump length. Normal diffusion, subdiffusion and superdiffusion can be realized within the present model. It is shown that the competition between long-tailed distribution and correlation of jump length and waiting time will lead to different diffusive behavior.     

Size matters: some stylized facts of the stock market revisited

We reanalyze high resolution data from the New York Stock Exchange and find a monotonic (but not power law) variation of the mean value per trade, the mean number of trades per minute and the mean trading activity with company capitalization. We show that the second moment of the traded value distribution is finite. Consequently, the Hurst exponents for the corresponding time series can be calculated. These are, however, non-universal: The persistence grows with larger capitalization and this results in a logarithmically increasing Hurst exponent. A similar trend is displayed by intertrade time intervals. Finally, we demonstrate that the distribution of the intertrade times is better described by a multiscaling ansatz than by simple gap scaling.