Ralf Metzler & Joseph
Klafter talk with ScienceWatch.com and answer a
few questions about this month's Emerging Research Front
Paper in the field of Mathematics.

Article: The restaurant at the end of the random
walk: recent developments in the description of anomalous
transport by fractional dynamics
Authors: Metzler,
R;Klafter, J
Journal: J PHYS-A-MATH GEN, 37 (31): R161-R208 AUG 6
2004
Addresses: NORDITA, Blegdamsvej 17, DK-2100 Copenhagen,
Denmark.
NORDITA, DK-2100 Copenhagen, Denmark.
Tel Aviv Univ, Sch Chem, IL-69978 Tel Aviv, Israel.

Why do you think your paper is highly
cited?

The paper represents an up-to-date introduction to the mathematics of
diffusion in complex systems involving Lévy stable laws which, in
turn, are connected to the generalized central limit theorem. The paper
summarizes a large body of experimental findings of anomalous diffusion
phenomena, and is therefore useful both for theorists and experimentalists
alike.

The paper is a follow-up on our first review article on the fractional
Fokker-Planck equation: Metzler, R; Klafter, J, "The random walk's guide to
anomalous diffusion: a fractional dynamics approach," Phys Rep-Rev Sect
Phys Lett 339:1-77, DEC 2000. The article was an Essential Science
Indicators^{SM}New Hot Paper selection in May 2003.

It collects the recent developments in the fast growing field of anomalous
transport processes, covering subdiffusion and emphasizing superdiffusion
phenomena. The collected evidence demonstrates that anomalous diffusion is
ubiquitous and reaches into a large variety of fields, such as physics,
chemistry, geophysics, astrophysics, but also the biological, financial,
and sociological fields.

Does it describe a new discovery, methodology, or
synthesis of knowledge?

The fractional Fokker-Planck equation in the form presented in R. Metzler,
et al., "Anomalous diffusion and relaxation close to thermal
equilibrium: A fractional Fokker-Planck equation approach," Phys. Rev.
Lett. 82: 3563-67, 1999; and in R. Metzler, et al., "Deriving
fractional Fokker-Planck equations from a generalized master equation,"
Europhys. Lett. 46: 431-36, 1999; is a general framework for
subdiffusion and superdiffusion in external force fields.

Coauthor
Joseph Klafter

Much as the regular Fokker-Planck equation, it provides a very general
description for random renewal processes with inherent, long-tailed memory.
It is connected with the Mittag-Leffler relaxation pattern identified in an
increasing number of complex systems, ranging from nano and microscales to
geophysics scales.

We also report on the space-fractional Fokker-Planck equation that
describes random walk processes in external force fields governed by a
long-tailed distribution of jump lengths with diverging variance. Such
Lévy flights arise naturally in jump models on annealed polymer
chains with intersegmental jumps and serve as models for large amplitude
noise.

In this paper we especially present recent findings on how to tame
Lévy flights: In steeper than harmonic potentials the resulting
process possesses finite variance and other interesting properties.

Moreover, in this paper we covered recent results on the first passage
behavior in both sub- and superdiffusive regimes. This paper is therefore a
guidebook for scientists working on transport phenomena within complex
systems.

Would you summarize the significance of your paper in
layman’s terms?

The classical drunken sailor's walk—after Karl Pearson
(1857–1936), English mathematician, biometrician, and
statistician—describes the trajectory of a walker moving in a random
direction with each step. The steps occur at regular time intervals. Such a
process leads to normal diffusion in the continuum limit, governed by the
famed Gaussian form of the probability density to find the walker at some
position at a given time.

The situation changes drastically when we introduce variable pauses between
successive steps. For instance, the already drunken sailor visits
additional pubs on his way. Assume that some of these pubs have such a good
beer that the sailor spends exceedingly long times in them. Then he most
likely will not make it back to his ship before its departure.

In nature, similar phenomena may occur. Imagine a tracer chemical, e.g.,
chloride, dissolved in the rainwater coming down on a catchment. The tracer
may get trapped inside channels off the main water artery. If these side
channels are very long, the transport of the tracer is governed by the
power-law return to the main artery. This causes a wide, power-law
distribution of waiting times with a diverging mean waiting time. If there
is an additional external force acting on the particle along the main
artery, its motion is described by the time-fractional Fokker-Planck
equation.

Lévy flights, i.e., random walk processes with power-law
distribution of jump lengths but typical waiting time between jumps, are
similarly described as a variant of the fractional Fokker-Planck equation.

How did you become involved in this research and were
any particular problems encountered along the way?

RM became involved in fractional differential equations during his thesis
work at the University of Ulm. When he came to Tel Aviv University to join
JK, who had an interest in anomalous diffusion and Lévy flights, we
realized that these equations are ideally suited to describe continuous
time random walk processes with power-law waiting time or jump length
distributions, in the presence of external force fields.

When one needs to solve a regular diffusion process in a force field,
mostly one would start with the Fokker-Planck equation. The fractional
Fokker-Planck equation is the starting point at exactly the same level for
particles performing anomalous diffusion. A challenge is how to phrase such
an equation if we are dealing with spatiotemporally coupled Lévy
walks.

Where do you see your research leading in the
future?

Currently, scientists in the field are dealing with the subtleties of such
anomalous diffusion properties when we calculate time averages instead of
the common ensemble averages. In fact, for a subdiffusion process described
by the fractional Fokker-Planck equation, ergodicity is broken in the sense
that the ensemble and time averages are no longer the same, caused by the
ageing properties of the underlying process. This also leads to the
question how one may better analyze measured single-particle trajectories.

Another open question is that of molecular crowding in biological cells:
what mechanism exactly causes the observed subdiffusion?

Do you foresee any social or political implications for
your research?

Realizing that diverging time scales govern certain processes should
eventually change the mindset within various fields. Prime examples are
that of groundwater studies and their related implications. Thus, spillage
of chemicals that eventually get into the groundwater will pose a long-term
challenge, as the chemicals are not washed out quickly, but a good portion
stays in the aquifer for considerable periods of time. Over decades, this
may compromise the water quality.

Lévy flights and walks have found their way to describe social
phenomena such as human mobility and the spread of epidemics. For instance,
the high connectivity of air traffic makes it possible to spread diseases
over continents within a time scale of only several days.

In the future, a better understanding of these related dynamics will help
to develop better measures against the spreading of diseases. Similarly,
better quantitative understanding of human mobility behavior will make it
possible to improve the design of the infrastructure of the future.

Ralf Metzler, Ph.D.
Professor
Department of Physics
Technical University of Munich
Garching, Germany Web |
Web

Joseph Klafter, Ph.D.
Professor
School of Chemistry
Tel Aviv University
Tel Aviv, Israel Web