First steps in random walks: from tools to applications

161 Pages·2015·0.94 MB·English

by  Klafter, J.;Sokolov, Igor M

First Steps in Random Walks
From Tools to Applications
J. Klafter and I.M. Sokolov
1
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Klafter,J.(Joseph)
Firststepsinrandomwalks:fromtoolstoapplications/J.Klafter,I.M.Sokolov.
p. cm.
ISBN 978–0–19–923486–8(hardback:acid-freepaper)
1. Randomwalks(Mathematics) I. Sokolov,IgorM.,1958- II. Title.
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ISBN 978–0–19–923486–8
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Preface
Thename“randomwalk”fortheproblemofdescribingthedisplacementofapointin
a sequence of independent random steps appeared when Karl Pearson posed in 1905
a question to the readers of Nature: “Can any of your readers refer me to a work
wherein I should find a solution of the following problem, or failing the knowledge of
any existing solution provide me with an original one? I should be extremely grateful
for aid in the matter. A man starts from a point O and walks l yards in a straight
line; he then turns through any angle whatever and walks anotherl yards in a second
straight line. He repeats this process n times. I require the probability that after n
stretches he is at a distance between r and r+δr from his starting point O.”
ThesameyearPearsonformulatedtheproblem,asimilarproblemwasformulated
inadifferentcontextbyAlbertEinsteininhisfamousarticle“Themotionofelements
suspended in static liquids as claimed in the molecular kinetic theory of heat,” which
formed part of his PhD work. An independent discussion based on a combinatorial
approach was published by Marian Smoluchowski a year later. However, the problem
of a random walk was posed even earlier by Louis Bachelier in his thesis devoted to
the theory of financial speculations in 1900.
The history of the random walk shows how versatile the model is: Pearson’s
questionwasmotivatedbymodelingthemotionofanimals(notmen,butmosquitoes!),
aproblemheworkedoninthosedays.Einstein’sworkwasclearlypertinenttophysics,
and Bachelier’s to economics (although Bachelier got his PhD in physics for it!).
Nowadays, the theory of random walks gives an approach that has proved useful in
physicsandchemistry(diffusionindifferentmedia,reactions,mixinginhydrodynamic
flows),economics,biology(fromthemotionofanimalstothatofsubcellularstructures
in the crowded environment inside cells), and many other disciplines. Corresponding
to a clear geometric and probabilistic model, the random-walk approach is extremely
powerful,andcanservenotonlyasamodelofsimplediffusion,butasamodelformany
complex sub- and superdiffusive transport processes as well. This book discusses the
mainvariantsoftherandom-walkmodelsandoffersthemostimportantmathematical
tools for theoretical descriptions of random walks.
Some words about the book should be given here.
Level.Thelevelofthediscussionisappropriateforastudentstartingscientificwork
or for a scientist entering a new field in which the random-walk approaches are used.
Both authors have taught courses on the subject for some years for undergraduate
and graduate students in chemistry and physics. The mathematical level of the
discussion is therefore that which can be followed after completion of the usual
mathematical courses for natural scientists or engineers. Any nonstandard material
(such as generating functions or fractional derivatives) is discussed in detail and by
use of exercises. In the case of higher levels of mathematical difficulties (especially
wheretheresultsarenotusedrepeatedly,inwhichcaseitwouldbebeneficialtolearn
iv Preface
something new), we have omitted the derivation and simply stated the result, giving
a reference.
Boxes. The most important or the most striking results are given in boxes. These
may serve as a short summary of the corresponding discussion.
Exercises. Exercises often contain parts of those derivations that we have omitted
in the main discussion, or give additional results. We often refer to the results of the
exercises in subsequent chapters. In this case the solution to the exercise is explicitly
stated in the text. Some of the exercises were considered to be serious scientific
problems in their time. Since the way to a solution is not always trivial, we give
hints. If you can solve the exercises, you can really master the subject.
References.Thisisaself-containedtextbook,andthereforeveryfewreferencesare
giventotheoriginalworks.Wemostlyrefertomathematicalcompendia(e.g.,whenwe
needtoevaluateanintegralortosumaseries)andtomonographsoreasilyaccessible
review articles where additional information can be found. The references to original
works are given only when they contain additional material that is not discussed in
detail in the text. Additional material, or discussions of mathematical problems that
may arise in connection with the discussions in the corresponding chapters, are also
given in Further reading.
Finally, we the authors had fun writing the book. We hope you will have fun
using it.
Contents
Abbreviations vii
1 Characteristic functions 1
1.1 First example: A random walk on a one-dimensional lattice 2
1.2 More general considerations 4
1.3 The moments 6
1.4 Random walk as a process with independent increments 6
1.5 A pedestrian approach to the Central Limit Theorem 7
1.6 When the Central Limit Theorem breaks down 10
1.7 Random walks in higher dimensions 13
2 Generating functions and applications 17
2.1 Definitions and properties 17
2.2 Tauberian theorems 19
2.3 Application to random walks: The first-passage and return
probabilities 21
2.4 Mean number of distinct visited sites 27
2.5 Sparre Andersen theorem 29
3 Continuous-time random walks 36
3.1 Waiting-time distributions 36
3.2 Transforming steps into time 39
3.3 Moments of displacement in CTRW 42
3.4 Power-law waiting-time distributions 43
3.5 Mean number of steps, MSD, and probability of being at the origin 49
3.6 Other characteristic properties of heavy-tailed CTRW 51
4 CTRW and aging phenomena 54
4.1 When the process ages 54
4.2 Forward waiting time 55
4.3 PDF of the walker’s positions 60
4.4 Moving time averages 62
4.5 Response to the time-dependent field 65
5 Master equations 68
5.1 A heuristic derivation of the generalized master equation 70
5.2 A note on time-dependent transition probabilities 74
5.3 Relation between the solutions to the generalized and the
customary master equations 75
5.4 Generalized Fokker–Planck and diffusion equations 77
vi Contents
6 FractionaldiffusionandFokker–Planckequationsforsubdiffusion 80
6.1 Riemann–Liouville and Weyl derivatives 80
6.2 Gru¨nwald–Letnikov representation 83
6.3 Fractional diffusion equation 84
6.4 Eigenfunction expansion 89
6.5 Subordination and the forms of the PDFs 92
7 L´evy flights 97
7.1 General L´evy distributions 98
7.2 Space-fractional diffusion equation for L´evy flights 102
7.3 Leapover 104
7.4 Simulation of L´evy distributions 106
8 Coupled CTRW and L´evy walks 110
8.1 Space–time coupled CTRWs 110
8.2 L´evy walks 114
8.3 L´evy walk interrupted by rests 119
9 Simple reactions: A+B→B 123
9.1 Configurational averaging 123
9.2 A target problem 125
9.3 Trapping problem 127
9.4 Asymptotics of trapping kinetics in one dimension 129
9.5 Trapping in higher dimensions 132
10 Random walks on percolation structures 135
10.1 Some facts about percolation 137
10.2 Fractals 139
10.3 Random walks on fractals 141
10.4 Calculating the spectral dimension 143
10.5 Using the spectral dimension 145
10.6 The role of finite clusters 147
Index 151
Abbreviations
CLT Central Limit Theorem
CTRW continuous-time random walks
GCLT generalized Central Limit Theorem
GME generalized master equation
MSD mean squared displacement
PDF probability density function
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1
Characteristic functions
“Life is the sum of trifling motions.”
Joseph Brodsky
In this chapter we introduce the characteristic function, a tool that plays a central
role in the mathematical description of random walks. When Karl Pearson posed in
1905hisquestion(reproducedinthePreface)tothereadersofNature hewasunaware
of how simply the problem could be formulated in terms of characteristic functions.
A trajectory of the Pearson’s random walk is shown in Fig. 1.1.
After getting acquainted with the powerful instrument of characteristic func-
tions, you should be able to answer Pearson’s question by performing a short
calculation.
12
8
4
0
−4
−9 −5 −1 3 7
Fig. 1.1 Pearson’s random walk of 200 steps of unit length.