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: benichou@[EMAIL PROTECTED]
).
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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.
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