First-passage times in complex scale-invariant media

[Roger Bagula]

http://www.nature.com/nature/journal/v450/n7166/full/nature06201.html
http://arxiv.org/abs/0711.0682
http://arxiv.org/pdf/0711.0682

First-passage times in complex scale-invariant media

S. Condamin1, O. Bénichou1, V. Tejedor1, R. Voituriez1 & J. Klafter2

1. Université Pierre et Marie Curie-Paris 6, Laboratoire de Physique
Théorique de la Matière Condensée, UMR CNRS 7600, case 121, 4 Place
Jussieu, 75005 Paris, France
2. School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel

Correspondence to: O. Bénichou1 Correspondence and requests for
materials should be addressed to O.B. (Email: ***@lptmc.jussieu.fr).
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Abstract

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 decade1. The
importance of FPTs originates from the crucial role played by first
encounter properties in various real situations, including transport in
disordered media2, 3, neuron firing dynamics4, spreading of diseases5 or
target search processes6, 7, 8, 9. 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 media1. 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 media10, fractals3, anomalous
diffusion11 and scale-free networks12.

[Roger Bagula]

--- In ***@yahoogroups.com, "Robert Karl Stonjek"
<***@...> wrote:

Fate might not be so unpredictable after all, study suggests

Why does it take so long for soul mates to find each other? How does
disease spread through a person's body? When will the next computer
virus attack your hard-drive?

A new theory published last month in Nature on the statistical concept
of "First Passage Time," or FPT, may provide the key to answering at
least a few of these questions, says theory co-author Prof. Joseph
Klafter from Tel Aviv University's School of Chemistry. And the answers
may lead to breakthroughs in medicine, mathematics, the environment, and
elsewhere.

Prof. Klafter and his colleagues from the University of Pierre & Marie
Curie in Paris (where he has been visiting professor) are the first to
have developed an analytical model that calculates the average arrival
time - the mean FPT - of a randomly-moving object in a complex environment.

Understanding how randomly-moving objects arrive at a certain
destination is no secret to scientists today. But no theory, until now,
could predict the time it would take for an object to move between given
addresses in a complex environment, like through the human body or the
World Wide Web. Previous models only explained the passage of time when
the event occurred in a homogenous environment, like in a vacuum or in a
glass of water.

And in some instances, such as the movement of cancer cells in the human
body, time is of the essence. The concept can best be understood by the
question: How long will it take for a random walker to reach a certain
destination"

Scientists from different backgrounds have studied and researched the
predictability of FPT for decades. "Our new theory is exciting because
it can be applied to a wide range of concepts in nature and
mathematics," explains Prof. Klafter, the Heinemann Chair of Physical
Chemistry at Tel Aviv University. "It can be used by biologists, by
ecologists, and even help computer scientists predict when the next big
virus will hit their computer."

When Prof. Klafter and his colleagues published their theory in Nature
on November 1, they sparked interest from around the world - especially
among biophysicists, who are looking for models to understand how long
it takes for molecules to arrive at certain points in biological cells.

And although it will take months, maybe even years, for real-life
experiments to prove the validity of this new theory, Prof. Klafter is
looking forward to the results.

"I've received responses from researchers who are interested in using
this model to analyze enzymes in cells," says Prof. Klafter. "Enzymes
are important for controlling functions in the body. If a biologist can
estimate the FPT of a certain enzyme (at the place where this molecule
reacts), then perhaps one could interfere with or manipulate the system
to help prevent a disease or make a bodily function more efficient."

He adds, "This theory can be applied to anything that moves randomly. It
can be used for predicting when an enzyme will reach a target cell, how
long a hungry animal will forage for food when food when is scarce - or
even how viruses spread through the Internet."

Source: American Friends of Tel Aviv University
http://www.physorg.com/news115917438.html

Posted by
Robert Karl Stonjek