Joseph Klafter

The random walk's guide to anomalous diffusion: a fractional dynamics approach

Abstract Fractional kinetic equations of the diffusion, diffusion–advection, and Fokker–Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have come of age as a complementary tool in the description of anomalous transport processes.

The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

An LED light source includes a rectangular LED chip and a transparent resin package enclosing the LED chip. The resin package is provided with a lens portion for directing light emitted from the LED chip to the outside of the resin package. The LED chip includes two adjacent side surfaces oriented toward the lens portion.

Beyond Brownian Motion

Newtonian physics began with an attempt to make precise predictions about natural phenomena, predictions that could be accurately checked by observation and experiment. The goal was to understand nature as a deterministic, “clockwork” universe. The application of probability distributions to physics developed much more slowly. Early uses of probability arguments focused on distributions with well‐defined means and variances. The prime example was the Gaussian law of errors, in which the mean traditionally represented the most probable value from a series of repeated measurements of a fixed quantity, and the variance was related to the uncertainty of those measurements.

The nonlinear nature of friction.

Tribology is the study of adhesion, friction, lubrication and wear of surfaces in relative motion. It remains as important today as it was in ancient times, arising in the fields of physics, chemistry, geology, biology and engineering. The more we learn about tribology the more complex it appears. Nevertheless, recent experiments coupled to theoretical modelling have made great advances in unifying apparently diverse phenomena and revealed many subtle and often non-intuitive aspects of matter in motion, which stem from the nonlinear nature of the problem.

Boundary value problems for fractional diffusion equations

The fractional diffusion equation is solved for different boundary value problems, these being absorbing and reflecting boundaries in half-space and in a box. Thereby, the method of images and the Fourier–Laplace transformation technique are employed. The separation of variables is studied for a fractional diffusion equation with a potential term, describing a generalisation of an escape problem through a fluctuating bottleneck. The results lead to a further understanding of the fractional framework in the description of complex systems which exhibit anomalous diffusion.

First-passage times in complex scale invariant media

How long does it take a random walker to reach a given target point? This quantity, known as a first-passage time (FPT), has led to a growing number of theoretical investigations over the past decade. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media, neuron firing dynamics, spreading of diseases or target search processes. Most methods of determining FPT properties in confining domains have been limited to effectively one-dimensional geometries, or to higher spatial dimensions only in homogeneous media. Here we develop a general theory that allows accurate evaluation of the mean FPT in complex media. Our analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source–target distance. The analysis is applicable to a broad range of stochastic processes characterized by length-scale-invariant properties. Our theoretical predictions are confirmed by numerical simulations for several representative models of disordered media, fractals, anomalous diffusion and scale-free networks.

Fractal behavior in trapping and reaction

We study trapping and reaction processes on fractals and compare the direct reaction with acceptors vs the multistep migration of the excitation. For the direct mechanism both exact and approximate expressions for the survival follow. For migration the trapping probability is determined from the number of distinct sites visited. We conclude that for each mechanism a different dimension of the fractal is decisive.

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 Levys 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.

From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion

Einsteins 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.

Dynamics of ionic solvation

Dynamics of solvation in simple polar liquids is studied within the framework of the mean spherical approximation. Exact results are derived for the Born solvation energy and for the correlation function for the solvation time of an instantaneously formed ion (or dipole) in a polar solvent. The results are in qualitative agreement with the recent approximate treatment by P. Wolynes [J. Chem. Phys. 86, 5133 (1987)]. Implications of the results for the solvation dynamics of dipoles and of excess electrons in polar solvents are considered.