Aspects and applications of the random walk:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam u.a.
North-Holland
1994
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| Series: | Random materials and processes
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 361 S. graph. Darst. |
| ISBN: | 0444816062 |
Staff View
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| 245 | 1 | 0 | |a Aspects and applications of the random walk |c George H. Weiss |
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Record in the Search Index
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|---|---|
| adam_text | xi
CONTENTS
PREFACE ix
DEDICATION xv
CHAPTER 1: INTRODUCTORY COMMENTS 1
1. Introduction 1
2. Basic definitions 3
2a. The independent random variable 3
2b. The Markov process 5
3. Preview of the book 9
References 17
CHAPTER 2: THE UBIQUITOUS CHARACTERISTIC FUNCTION 19
1. Introduction 19
2. The characteristic function 21
3. Some examples 27
3a. The drunkard s walk 27
3b. A random walk with Gaussian step lengths 28
3c. A second random walk with no restriction on step size 28
3d. The Pearson random walk in two and three dimensions 29
3e. The persistent random walk 33
4. General properties of the characteristic function 36
4a. The characteristic function as a moment generating function 36
4b. Cumulants 37
4c. Non integer and negative moments 40
4d. Some analytic properties of the characteristic function 42
5. The continuous time random walk (CTRW) 44
References 53
CHAPTER 3: ASYMPTOTICS AND THE DIFFUSION LIMIT 55
1. Introduction 55
2. Approximations based on the central limit theorem 57
3. The saddlepoint approximation 65
3a. The basic technique 65
3b. Some examples 68
3c. The saddlepoint approximation for power series 72
4. The diffusion approximation 75
4a. Derivation of the elementary diffusion equation 75
4b. Random walks with longer range jumps and the diffusion equation ..78
4c. Diffusion on a non homogeneous substrate (the Smoluchowski
equation) 80
xii CONTENTS
4d. The diffusion approximation for the persistent random walk 81
5. Abelian and Tauberian theorems 86
5a. Definition of these theorems 86
5b. Proof of an Abelian theorem 88
5c. Tauberian theorems 91
6. The CTRW revisited 95
6a. Pausing time densities with finite moments 95
6b. The symmetric CTRW with a pausing time density of stable
law form 100
6c. The asymmetric CTRW 107
7. Summary 112
References 112
CHAPTER 4: LATTICE WALKS 115
1. Introduction 115
2. The Polya problem and some generalizations 116
2a. The probability of return to the origin 116
2b. The probability of reaching a particular lattice site 120
2c. The Polya problem in the absence of moments 121
2d. The average return time for recurrent walks 122
2e. The Polya problem for k( l) random walkers 123
2f. The expected number of visits to a given site 128
2g. Expected number of sites visited k times in n steps 129
3. The number of distinct sites visited during an n step walk 131
3a. Random walks in discrete time 131
3b. The expected number of distinct sites visited by a CTRW 135
3c. The average number of distinct sites in a given set visited by
n step walk 136
3d. The expected number of sites visited by N independent
random walkers 138
4. The occupancy of a set of points 142
4a. Occupancy of a set on a lattice 142
4b. Occupancy of a set: the continuum version 148
5. Random walks on periodic lattices 152
5a. Fundamental properties 152
5b. First passage times to a point on a finite lattice 157
References 160
CHAPTER 5: BOUNDARIES AND CONSTRAINTS 163
1. Introduction 163
2. Derivation of boundary conditions 165
2a. Boundary conditions for the simplest random walk and for the
diffusion equation 165
2b. The reflecting boundary 166
2c. Reflecting and partially absorbing boundaries 168
2d. Boundary conditions for the persistent random walk 169
CONTENTS xiii
3. Some examples 170
3a. The diffusion process in one dimension with single trapping,
reflecting, or partially reflecting points 170
3b. The survival time and associated moments for a single trap 173
3c. The nearest neighbor problem for the diffusion process 175
3d. The diffusion process on a line surrounded by two trapping points . 177
3e. An alternative form of the solution 180
4. Wald s identity 182
5. Maximum excursions 188
5a. Random walks with a finite average displacement 188
5b. A random walk with an infinite mean jump size 192
5c. Properties of the maximum excursion for a CTRW 194
5d. The maximum excursion in three dimensions 195
6. The span(s) of a random walk 198
6a. Fundamental formalism 198
6b. The span for stable law random walks 203
6c. The span of a CTRW 206
6d. The asymmetry of the symmetric one dimensional random walks... 207
7. An outline of the trapping problem 212
7a. The Rosenstock approximation 212
7b. Trapping in one dimension 215
7c. A heuristic derivation of the Donsker Varadhan approximation 220
8. A constrained random walk 221
Appendix: A heuristic proof of the Poisson transformation 228
References 229
CHAPTER 6: MULTISTATE RANDOM WALKS 233
1. Introduction 233
1 2. Basic formalism 234
2a. Specific assumptions defining the multistate random walk 234
2b. Formal solution of the model 236
3. The two state random walk with applications 239
3a. Basic formalism and asymptotics 239
3b. Asymptotic properties of the Lennard Jones model 239
3c. Asymptotic properties of the moments 243
3d. The telegrapher s equation and some generalizations 246
3e. More general two state random walks 249
3f. Applications of the two state random walk 253
3g. Applications to the theory of chromatographic processes 255
3h. Some properties of the multistate random walk on a lattice 263
References 267
CHAPTER 7: SELECTED APPLICATIONS 269
1. Introduction 269
2. Applications of random walks in crystallography 271
2a. Basic ideas 271
xiv CONTENTS
2b. Exact representations of the probability densities for intensity
statistics 279
2c. Direct methods in the solution of the phase problem 281
3. Models for transport in a disordered medium 287
3a. Introduction 287
3b. A solvable hydrodynamic model 288
3b. Two models of disorder based on the CTRW 297
3c. The Scher Lax and Scher Montroll models 297
3d. Comb models for diffusion on an infinite percolation cluster 309
3e. Effective medium theories 324
3f. A perturbation expansion 330
4. The reptation model 335
5. A model for photon migration in a turbid medium 341
5a. Basic assumptions 341
5b. Fundamental solutions to the evolution equation 344
5c. Path length of a photon in the tissue 347
5d. Measures of the depth of penetration 348
References 357
INDEX 361
|
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| institution | BVB |
| isbn | 0444816062 |
| language | English |
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| spelling | Weiss, George H. Verfasser aut Aspects and applications of the random walk George H. Weiss Amsterdam u.a. North-Holland 1994 XIV, 361 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Random materials and processes Marche aléatoire Processus stochastique Processus stochastiques ram Random walks (statistiek) gtt Random walks (Mathematics) Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006401941&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Weiss, George H. Aspects and applications of the random walk Marche aléatoire Processus stochastique Processus stochastiques ram Random walks (statistiek) gtt Random walks (Mathematics) Irrfahrtsproblem (DE-588)4162442-7 gnd |
| subject_GND | (DE-588)4162442-7 |
| title | Aspects and applications of the random walk |
| title_auth | Aspects and applications of the random walk |
| title_exact_search | Aspects and applications of the random walk |
| title_full | Aspects and applications of the random walk George H. Weiss |
| title_fullStr | Aspects and applications of the random walk George H. Weiss |
| title_full_unstemmed | Aspects and applications of the random walk George H. Weiss |
| title_short | Aspects and applications of the random walk |
| title_sort | aspects and applications of the random walk |
| topic | Marche aléatoire Processus stochastique Processus stochastiques ram Random walks (statistiek) gtt Random walks (Mathematics) Irrfahrtsproblem (DE-588)4162442-7 gnd |
| topic_facet | Marche aléatoire Processus stochastique Processus stochastiques Random walks (statistiek) Random walks (Mathematics) Irrfahrtsproblem |
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