First-passage Phenomena And Their Applications

Ralf Metzler, Gleb Oshanin, Sidney Redner

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814590303
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

The book contains review articles on recent advances in first-passage phenomena and applications contributed by leading international experts. It is intended for graduate students and researchers who are interested in learning about this intriguing and important topic.

Contents:

• Arrival Statistics and Exploration Properties of Mortal Walkers (S B Yuste, E Abad and K Lindenberg)
• First Passage of a Randomly Accelerated Particle (T W Burkhardt)
• First Passage Problems in Anomalous Diffusion (A Rosso and A Zoia)
• First-Passage Times of Intermittent Random Walks (O Bénichou and R Voituriez)
• First-Passage Phenomena on Finite Inhomogeneous Networks (E Agliari and D Cassi)
• Effective Spectral Dimension in Scale-Free Networks (S Hwang, D-S Lee and B Kahng)
• First-Passage Statistics for Random Walks in Bounded Domains (R Voituriez and O Bénichou)
• First Passage Behavior of Multi-Dimensional Fractional Brownian Motion and Application to Reaction Phenomena (J-H Jeon, A V Chechkin and R Metzler)
• Trajectory-to-Trajectory Fluctuations in First-Passage Phenomena in Bounded Domains (T G Mattos, C Mejía-Monasterio, R Metzler, G Oshanin and G Schehr)
• Exact Record and Order Statists of Random Walk via First-Passage Ideas (G Schehr and S N Majumdar)
• First Passage in a Conical Geometry and Ordering of Brownian Particles (E Ben-Naim and P L Krapivsky)
• First Passage Time Problems in Biophysical Jump Processes with Fast Kinetics (P C Bressloff and J M Newby)
• First Passage Problems in Biology (T Chou and M R D'Orsogna)
• The Effect of Detection Mechanisms on Spatial Search and Foraging (D Campos and V Méndez)
• Search in Random Media with Lévy Flights (E Gelenbe and O H Abdelrahman)
• Exit Strategies: Visual Search and the Quitting Time Problem (T S Horowitz)
• Statistical Physics of Evolutionary Trajectories on Fitness Landscapes (M Manhart and A V Morozov)
• Some Applications of First-Passage Ideas to Finance (R Chicheportiche and J-P Bouchaud)
• First-Passage and Extremes in Socio-Economic Systems (J Masoliver and J Perelló)
• Transport and the First-Passage Time Problem with Application to Cold Atoms in Optical Traps (E Barkai and D A Kessler)
• The Excursion Set Theory in Cosmology (M Maggiore and A Riottoo)
• Self-Organized Escape Processes of Linear Chains in Nonlinear Potentials (T Gross, D Hennig and L Schimansky-Geier)
• Efficient Monte Carlo Methods for Simulating Diffusion-Reaction Processes in Complex Systems (D S Grebenkov)

Readership: Researchers in stochastic processes, statistical physics, and mathematical physics.
Key Features:

• Comprehensive update of the classical book by Sidney Redner
• Applications to wide-ranging and active fields of research
• Well-known authors in the field

Chapter 1

Arrival Statistics and Exploration Properties of Mortal
Walkers

Santos B. Yuste

Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
[email protected]

Enrique Abad

Departamento de Física Aplicada, Centro Universitario de Mérida, Universidad de Extremadura, E-06800 Mérida, Spain
[email protected]

Katja Lindenberg

Department of Chemistry and Biochemistry and BioCircuits Institute, University of California San Diego, La Jolla, CA 92093-0340, USA
[email protected]

We study some of the salient features of the arrival statistics and exploration properties of mortal random walkers, that is, walkers that may die as they move, or as they wait to move. Such evanescence or death events have profound consequences for quantities such as the number of distinct sites visited which are relevant for the computation of encounter-controlled rates in chemical kinetics. We exploit the observation that well-known methods developed decades ago for immortal walkers are widely applicable to mortal walkers. The particular cases of exponential and power-law evanescence are considered in detail. Finally, we discuss the relevance of our results to the target problem with mortal traps and a particular application thereof, namely, the defect diffusion model. Evanescence of defects is postulated as a possible complementary contribution or perhaps even an alternative to anomalous diffusion to explain observed stretched exponential relaxation behavior.

1. Introduction

Random walks play a central role in statistical physics as elementary models describing stochastic transport in a large variety of experimental systems [1]. While there is a large body of literature dealing with the first-passage and exploration behavior of walkers that do not undergo reactions or transformations, considerably less attention has been devoted to the corresponding properties of reactive walkers. In particular, work involving exploration properties of so-called mortal or evanescent walkers, that is, walkers that die in the course of their motion, is surprisingly scarce despite the ubiquitous occurrence of such death processes in many physical, chemical, and biological scenarios.

Ad-hoc models for particular experimental situations have previously been implemented, e.g., to deal with photon scattering and absorption in a tissue [2]. And yet, with very few exceptions (see e.g. [2, 3] or subsection 3.2.4 in [4]) the observation that the effect of elementary death processes on the exploration properties of random walkers can be investigated using arguments similar to those used for immortal walkers seems until recently to have gone largely unnoticed [5]. The aim of the present work is to introduce in a pedagogical way an overarching theoretical framework allowing one to obtain the most interesting properties of mortal walkers. In view of the completely new physics introduced by death processes in the transport behavior and its potential relevance for experiments such as for instance the defect relaxation system discussed in Sec. 4, this endeavor seems very timely.

In what follows we present in a more general framework some recent results first given in [5] in a succinct fashion. More specifically, we shall focus on properties characterizing the arrival statistics of random walkers in discrete time, such as the average number of distinct sites visited after a certain number of steps on perfect lattices of all dimensions and other related quantities. In the continuum time limit one can establish a relationship between these quantities and the number of distinct sites visited up to a given time (rather than step number), a key quantity for the computation of reaction rates in chemical kinetics.

Mortal random walkers have a probability of dying as they walk. This means that their density ρ is not a constant as it is in an ordinary random walk but is instead a decaying function of time, ρ(t), or of step number, ρ(n). One can think of this decay as arising from a process that removes the walkers or transforms them into an inert species This might be the result of...

Table of contents

• Cover Page
• Title Page
• Copyright
• Preface
• Contents
• 1. Arrival Statistics and Exploration Properties of Mortal Walkers
• 2. First Passage of a Randomly Accelerated Particle
• 3. First Passage Problems in Anomalous Diffusion
• 4. First-Passage Times of Intermittent Random Walks
• 5. First-Passage Phenomena on Finite Inhomogeneous Networks
• 6. Effective Spectral Dimension in Scale-Free Networks
• 7. First-Passage Statistics for Random Walks in Bounded Domains
• 8. First Passage Behavior of Multi-Dimensional Fractional Brownian Motion and Application to Reaction Phenomena
• 9. Trajectory-to-Trajectory Fluctuations in First-Passage Phenomena in Bounded Domains
• 10. Exact Record and Order Statistics of Random Walks via First-Passage Ideas
• 11. First Passage in Conical Geometry and Ordering of Brownian Particles
• 12. First Passage Time Problems in Biophysical Jump Processes with Fast Kinetics
• 13. First Passage Problems in Biology
• 14. The Effect of Detection Mechanisms on Spatial Search and Foraging
• 15. Search in Random Media with L´evy Flights
• 16. Exit Strategies: Visual Search and the Quitting Time Problem
• 17. Statistical Physics of Evolutionary Trajectories on Fitness Landscapes
• 18. Some Applications of First-Passage Ideas to Finance
• 19. First-Passage and Extremes in Socio-Economic Systems
• 20. Transport and the First Passage Time Problem with Application to Cold Atoms in Optical Traps
• 21. The Excursion Set Theory in Cosmology
• 22. Self-Organized Escape Processes of Linear Chains in Nonlinear Potentials
• 23. Efficient Monte Carlo Methods for Simulating Diffusion- Reaction Processes in Complex Systems