Michael F. Shlesinger

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.

Time‐Scale Invariance in Transport and Relaxation

An early theme in probability was calculating the fair ante for various games of chance. Nicolas Bernoulli introduced a seemingly innocent game, first published in 1713, that yielded a paradoxical result. The result has become known as the St. Petersburg paradox, because of an analysis written later by Daniel Bernoulli in the Commentary of the St. Petersburg Academy.

Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise : a tale of tails

In this report on examples of distribution functions with long tails we (a) show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function, (b) review several models that yield log-normal distributions, (c) show that log normal distributions may mimic 1/f noise over a certain range, and (d) present an amplification model to show how log-normal personal income distributions are transformed into inverse power (Pareto) distributions in the high income range.

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.

ON THE UBIQUITY OF 1/f NOISE

A generic mechanism for the ubiquitous phenomenon of 1/f noise is reviewed. This mechanism arises in random processes expressible as a product of several random variables. Under mild conditions this product form leads to the log-normal distribution which we show straightforwardly generates 1/f noise. Thus, 1/f noise is tied directly to a probability limit distribution. A second mechanism involving scaling is introduced to provide a natural crossover from log-normal to inverse power-law behavior and generates 1/fα noise instead of pure 1/f noise. Examples of these distributions and the transitions between them are drawn from such diverse areas as economics, scientific productivity, bronchial structure and cardiac activity.

Williams-watts dielectric relaxation: A fractal time stochastic process

Dielectric relaxation in amorphous materials is treated in a defect-diffusion model where relaxation occurs when a mobile defect, such as a vacancy, reaches a frozen-in dipole. The random motion of the defect is assumed to be governed by a fractal time stochastic process where the mean duration between defect movements is infinite. When there are many more defects than dipoles, the Williams-Watts decaying fractional exponential relaxation law is derived. The argument of the exponential is related to the number of distinct sites visited by the random walk of the defect. For the same reaction dynamics but with more traps than walkers, an algebraically decaying relaxation is found.

Mathematical physics: search research.

How does one best search for non-replenishable targets at unknown positions? An optimized search strategy could be applied to situations as diverse as animal foraging and time-sensitive rescue missions.

Lévy Walks Versus Lévy Flights

We explore the behavior of random walkers that fly instantaneously between successive sites, however distant, and those that must walk between these sites. The latter case is related to intermittent behavior in Joseph- son junctions and to turbulent diffusion.

Fractal Time and 1/f Noise in Complex Systems

The later part of the twentieth century seemed to be a period when science and mathematics were becoming more and more specialized. Remarkably, in the last decade, this trend has been reversed due to two major themes: nonlinear dynamics and fractals. The former has involved the discovery of generic universal behavior of nonlinear deterministic equations of motion, and the latter has involved a geometry (and dynamics on that geometry) of self-similar and self-afine objects. Both themes have been applied to an impressive array of interdisciplinary problems. A key to the success of these approaches has been their global rather than specific nature. Physicists, mathematicians, chemists, engineers, biologists, etc., have all by now constructed phase-space plots and calculated fractal dimensions. This paper is concerned with fractal behavior in time rather than in space. This concept has met with much success in the physics of amorphous materials and is just beginning to be introduced into the biological sciences. We will begin by discussing the notion of scaling that underlies fractal time. We then will define fractal time and mention its applications to transport, reaction, and relaxation in disordered materials. Finally, we will show how a limiting case of fractal time is related to the famous problem of l/f noise.

Generalized Vogel law for glass-forming liquids

A model for non-Arrhenius structural and dielectric relaxation in glass-forming materials is based on defect clustering in supercooled liquids. Relaxation in the cold liquid is highly hindered, and assumed to require the presence of a mobile defect to loosen the structure near it. A mild distribution of free-energy barriers impeding defect hopping can generate a wide distribution of waiting times between relaxation events. When the mean waiting time is longer than the time of an experiment, no characteristic time scale exists. This case directly yields the Kohlrausch-Williams-Watts (KWW) relaxation law. A free-energy mismatch between defect and nondefect regions produces a defect-defect attraction, which can lead to aggregation. This may occur in defect-rich “fragile” liquids which also exhibit Vogel kinetics. Defect aggregation and correlation in the “high-temperature” region above the critical consolute temperatureTc is described using the Ornstein-Zernike theory of critical fluctuations. For a defect correlation length divergence (T-Tc)-γ/2, a generalized Vogel law for the structural relaxation time τ results: τ=τ0exp[B./(T-Tc)1.5γ] In the mean-field limit (γ=1) this provides as good an account of dielectric and structural relaxation in glycerol,n-propanol, andi-butyl bromide as does the original Vogel law, and for the mixed salt KNO3−Ca(NO3)2 and B2O2 it also describes kinetics over their entire temperature ranges. A breakdown of the Vogel law in the immediate vicinity ofTg is avoided, and the need to invoke extra low-temperature mechanisms to explain an apparent “return to Arrhenius behavior” is removed.