Stochastic Equations through the Eye of the Physicist (eBook)

Basic Concepts, Exact Results and Asymptotic Approximations

Valery I. Klyatskin (Autor)

eBook Download: PDF

2005 | 1. Auflage
556 Seiten
Elsevier Science (Verlag)
978-0-08-045764-2 (ISBN)

Infos

"Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well known example of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the foundation for modern stochastic calculus and statistical physics. Other important examples include turbulent transport and diffusion of particle-tracers (pollutants), or continuous densities (''oil slicks''), wave propagation and scattering in randomly inhomogeneous media, for instance light or sound propagating in the turbulent atmosphere.

Such models naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields.

The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data.

This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated nonlinear functional of random fields and processes.

Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools.

Part II and III sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous examples.

Part IV takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering), wave propagation in disordered 2D and 3D media.

For the sake of reader I provide several appendixes (Part V) that give many technical mathematical details needed in the book.

For scientists dealing with stochastic dynamic systems in different areas, such as hydrodynamics, acoustics, radio wave physics, theoretical and mathematical physics, and applied mathematics

the theory of stochastic in terms of the functional analysis

Referencing those papers, which are used or discussed in this book and also recent review papers with extensive bibliography on the subject."

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Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well known example of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the foundation for modern stochastic calculus and statistical physics. Other important examples include turbulent transport and diffusion of particle-tracers (pollutants), or continuous densities (''oil slicks''), wave propagation and scattering in randomly inhomogeneous media, for instance light or sound propagating in the turbulent atmosphere. Such models naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields. The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data. This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated "e;nonlinear functional"e; of random fields and processes. Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools. Part II and III sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous examples. Part IV takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering), wave propagation in disordered 2D and 3D media. For the sake of reader I provide several appendixes (Part V) that give many technical mathematical details needed in the book. - For scientists dealing with stochastic dynamic systems in different areas, such as hydrodynamics, acoustics, radio wave physics, theoretical and mathematical physics, and applied mathematics- The theory of stochastic in terms of the functional analysis- Referencing those papers, which are used or discussed in this book and also recent review papers with extensive bibliography on the subject

Inhalt

Front Cover 1
Stochastic Equations Through The Eye of The Physicist 4
Copyright Page 5
Contents 10
Preface 8
Introduction 16
Part I: Dynamical description of stochastic systems 20
Chapter 1. Examples, basic problems, peculiar features of solutions 21
1.1 Ordinary differential equations: initial value problems 21
1.2 Linear ordinary differential equations: boundary-value problems 29
1.3 First-order partial differential equations 38
1.4 Partial differential equations of higher orders 46
1.5 Solution dependence on medium parameters and initial value 54
Chapter 2. Indicator function and Liouville equation 57
2.1 Ordinary differential equations 57
2.2 First-order partial differential equations 58
2.3 Higher-order partial differential equations 62
Part II: Stochastic equations 66
Chapter 3. Random quantities, processes, and fields 67
3.1 Random quantities and their characteristics 67
3.2 Random processes, fields, and their characteristics 71
3.3 Markovian processes 85
Chapter 4. Correlation splitting 96
4.1 General remarks 96
4.2 Gaussian process 98
4.3 Poisson process 100
4.4 Telegrapher's random process 101
4.5 Generalized telegrapher's random process 104
4.6 General-form Markovian processes 105
4.7 Delta-correlated random processes 108
Chapter 5. General approaches to analyzing stochastic dynamic systems 115
5.1 Ordinary differential equations 115
5.2 Partial differential equations 119
5.3 Stochastic integral equations (methods of quantum field theory) 123
5.4 Completely solvable stochastic dynamic systems 134
5.5 Delta-correlated fields and processes 149
Chapter 6. Stochastic equations with the Markovian fluctuations of parameters 169
6.1 Telegrapher's processes 170
6.2 Generalized telegrapher's process 179
6.3 Gaussian Markovian processes 187
6.4 Markovian processes with finite-dimensional phase space 191
6.5 Causal stochastic integral equations 193
Part III: Asymptotic and approximate methods for analyzing stochastic equations 202
Chapter 7. Gaussian random field delta-correlated in time (ordinary differential equations) 203
7.1 The Fokker–Planck equation 203
7.2 Transitional probability distributions 205
7.3 Applicability range of the Fokker–Planck equation 207
Chapter 8. Methods for solving and analyzing the Fokker-Planck equation 212
8.1 System of linear equations 212
8.2 Integral transformations 221
8.3 Steady-state solutions of the Fokker–Planck equation 223
8.4 Boundary-value problems for the Fokker-Planck equation (transfer phenomena) 228
8.5 Asymptotic and approximate methods of solving the Fokker-Plank equation 234
Chapter 9. Gaussian delta-correlated random field (causal integral equations) 241
9.1 Causal integral equation 242
9.2 Statistical averaging 242
Chapter 10. Diffusion approximation 246
10.1 General remarks 246
10.2 Dynamics of a particle 247
Part IV: Coherent phenomena in stochastic dynamic systems 252
Chapter 11. Passive tracer diffusion and clustering in random hydrodynamic flows 253
11.1 General remarks 253
11.2 Statistical description 259
11.3 Additional factors 279
Chapter 12. Wave localization in randomly layered media 297
12.1 General remarks 297
12.2 Statistics of scattered field at layer boundaries 301
12.3 Statistical description of a wavcficld in random medium 317
12.4 Eigenvalue and eigenfunction statistics 346
12.5 Multidimensional wave problems in layered random media 353
12.6 Two-layer model of the medium 365
Chapter 13. Wave propagation in random media 374
13.1 Method of stochastic equation 374
13.2 Geometrical optics approximation in randomly inhomogeneous media 398
13.3 Method of path integral 411
Chapter 14. Some problems of statistical hydrodynamics 432
14.1 Quasi-elastic properties of isotropic and stationary noncomprcssiblc turbulent media 433
14.2 Sound radiation by vortex motions 436
Part V: Appendixes 446
A. Variation (functional) derivatives 447
B. Fundamental solutions of wave problems in free space and layered media 452
B.1 Free space 452
B.2 Layered space 455
C. Imbedding method in boundary-value wave problems 458
C. 1 Boundary-value problems formulated in terms of ordinary differential equations 459
C.2 Stationary boundary-value wave problems 462
C.3 One-dimensional nonstationary boundary-value wave problem 517
Bibliography 532
Index 554

Produkt-Details

• Erscheint lt. Verlag: 20.5.2005
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Thermodynamik
• Themenwelt: Technik
• ISBN-10: 0-08-045764-9 / 0080457649
• ISBN-13: 978-0-08-045764-2 / 9780080457642