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E-raamat: Lectures on Dynamics of Stochastic Systems

(Russian Academy of Science, Russia)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 09-Sep-2010
  • Kirjastus: Elsevier Science Publishing Co Inc
  • Keel: eng
  • ISBN-13: 9780123849670
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Lectures on Dynamics of Stochastic SystemsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 09-Sep-2010
  • Kirjastus: Elsevier Science Publishing Co Inc
  • Keel: eng
  • ISBN-13: 9780123849670
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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. Models naturally render to statistical description, where random processes and fields express the input parameters and solutions. 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 book is a revised and more comprehensive version of Dynamics of Stochastic Systems. Part I provides an introduction to the topic. Part II is devoted to the general theory of statistical analysis of dynamic systems with fluctuating parameters described by differential and integral equations. Part III deals with the analysis of specific physical problems associated with coherent phenomena

  • A comprehensive update of Dynamics of Stochastic Systems
  • Develops mathematical tools of stochastic analysis and applies them to a wide range of physical models of particles, fluids and waves
  • Includes problems for the reader to solve


    • 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. Models naturally render to statistical description, where random processes and fields express the input parameters and solutions. The fundamental problem of stochastic dynamics is to identify the essential characteristics of the system (its state and evolution), and relate those to the input parameters of the system and initial data.

    This book is a revised and more comprehensive version of Dynamics of Stochastic Systems. Part I provides an introduction to the topic. Part II is devoted to the general theory of statistical analysis of dynamic systems with fluctuating parameters described by differential and integral equations. Part III deals with the analysis of specific physical problems associated with coherent phenomena.

    • A comprehensive update of Dynamics of Stochastic Systems
    • Develops mathematical tools of stochastic analysis and applies them to a wide range of physical models of particles, fluids and waves
    • Includes problems for the reader to solve

    Arvustused

    "...an extensively revised and more detailed version of the author's earlier book...a statistical description and analysis of dynamic systems that occur in physics and related areas." --MathSciNet, Lectures on Dynamics of Stochastic Systems"Taking into account opinions and wishes of readers about both the style of the text and the choice of specific problems, the aim of the book at this edition is simply to present the subject of its title sourced from the series of lectures that the author gave to scientific associates at the Institute of Calculus Mathematics, Russian Academy of Sciences. Each lecture is appended with problems for readers." --Zentralblatt MATH 2012-1233-93001

    Muu info

    Reviews the latest developments in stochastic dynamic systems covering areas such as acoustics, hydrodynamics, magneto-hydrodynamics, radiophysics, theoretical and mathematical physics and applied mathematics
    Preface ix
    Introduction xi
    Part I Dynamical Description of Stochastic Systems
    		1(86)
    1 Examples, Basic Problems, Peculiar Features of Solutions
    		3(50)
    1.1 Ordinary Differential Equations: Initial-Value Problems
    		3(17)
    1.1.1 Particles Under the Random Velocity Field
    		3(5)
    1.1.2 Particles Under Random Forces
    		8(2)
    1.1.3 The Hopping Phenomenon
    		10(8)
    1.1.4 Systems with Blow-Up Singularities
    		18(1)
    1.1.5 Oscillator with Randomly Varying Frequency (Stochastic Parametric Resonance)
    		19(1)
    1.2 Boundary-Value Problems for Linear Ordinary Differential Equations (Plane Waves in Layered Media)
    		20(4)
    1.3 Partial Differential Equations
    		24(26)
    1.3.1 Linear First-Order Partial Differential Equations
    		24(9)
    1.3.2 Quasilinear and Nonlinear First-Order Partial Differential Equations
    		33(6)
    1.3.3 Parabolic Equation of Quasioptics (Waves in Randomly Inhomogeneous Media)
    		39(3)
    1.3.4 Navier-Stokes Equation: Random Forces in Hydrodynamic Theory of Turbulence
    		42(8)
    Problem
    		50(3)
    2 Solution Dependence on Problem Type, Medium Parameters, and Initial Data
    		53(16)
    2.1 Functional Representation of Problem Solution
    		53(7)
    2.1.1 Variational (Functional) Derivatives
    		53(6)
    2.1.2 Principle of Dynamic Causality
    		59(1)
    2.2 Solution Dependence on Problem's Parameters
    		60(5)
    2.2.1 Solution Dependence on Initial Data
    		60(2)
    2.2.2 Imbedding Method for Boundary-Value Problems
    		62(3)
    Problems
    		65(4)
    3 Indicator Function and Liouville Equation
    		69(18)
    3.1 Ordinary Differential Equations
    		69(3)
    3.2 First-Order Partial Differential Equations
    		72(8)
    3.2.1 Linear Equations
    		72(5)
    3.2.2 Quasilinear Equations
    		77(2)
    3.2.3 General-Form Nonlinear Equations
    		79(1)
    3.3 Higher-Order Partial Differential Equations
    		80(5)
    3.3.1 Parabolic Equation of Quasi-Optics
    		80(3)
    3.3.2 Random Forces in Hydrodynamic Theory of Turbulence
    		83(2)
    Problems
    		85(2)
    Part II Statistical Description of Stochastic Systems
    		87(182)
    4 Random Quantities, Processes, and Fields
    		89(34)
    4.1 Random Quantities and their Characteristics
    		89(6)
    4.2 Random Processes, Fields, and their Characteristics
    		95(20)
    4.2.1 General Remarks
    		95(4)
    4.2.2 Statistical Topography of Random Processes and Fields
    		99(3)
    4.2.3 Gaussian Random Process
    		102(3)
    4.2.4 Gaussian Vector Random Field
    		105(3)
    4.2.5 Logarithmically Normal Random Process
    		108(2)
    4.2.6 Discontinuous Random Processes
    		110(5)
    4.3 Markovian Processes
    		115(4)
    4.3.1 General Properties
    		115(2)
    4.3.2 Characteristic Functional of the Markovian Process
    		117(2)
    Problems
    		119(4)
    5 Correlation Splitting
    		123(18)
    5.1 General Remarks
    		123(2)
    5.2 Gaussian Process
    		125(2)
    5.3 Poisson's Process
    		127(1)
    5.4 Telegrapher's Random Process
    		128(2)
    5.5 Delta-Correlated Random Processes
    		130(5)
    5.5.1 Asymptotic Meaning of Delta-Correlated Processes and Fields
    		133(2)
    Problems
    		135(6)
    6 General Approaches to Analyzing Stochastic Systems
    		141(42)
    6.1 Ordinary Differential Equations
    		141(3)
    6.2 Completely Solvable Stochastic Dynamic Systems
    		144(16)
    6.2.1 Ordinary Differential Equations
    		144(14)
    6.2.2 Partial Differential Equations
    		158(2)
    6.3 Delta-Correlated Fields and Processes
    		160(6)
    6.3.1 One-Dimensional Nonlinear Differential Equation
    		162(3)
    6.3.2 Linear Operator Equation
    		165(1)
    Problems
    		166(17)
    7 Stochastic Equations with the Markovian Fluctuations of Parameters
    		183(8)
    7.1 Telegrapher's Processes
    		184(3)
    7.2 Gaussian Markovian Processes
    		187(1)
    Problems
    		188(3)
    8 Approximations of Gaussian Random Field Delta-Correlated in Time
    		191(38)
    8.1 The Fokker-Planck Equation
    		191(3)
    8.2 Transition Probability Distributions
    		194(2)
    8.3 The Simplest Markovian Random Processes
    		196(15)
    8.3.1 Wiener Random Process
    		197(1)
    8.3.2 Wiener Random Process with Shear
    		197(3)
    8.3.3 Logarithmic-Normal Random Process
    		200(11)
    8.4 Applicability Range of the Fokker-Planck Equation
    		211(4)
    8.4.1 Langevin Equation
    		211(4)
    8.5 Causal Integral Equations
    		215(3)
    8.6 Diffusion Approximation
    		218(2)
    Problems
    		220(9)
    9 Methods for Solving and Analyzing the Fokker-Planck Equation
    		229(24)
    9.1 Integral Transformations
    		229(1)
    9.2 Steady-State Solutions of the Fokker-Planck Equation
    		230(12)
    9.2.1 One-Dimensional Nonlinear Differential Equation
    		231(1)
    9.2.2 Hamiltonian Systems
    		232(2)
    9.2.3 Systems of Hydrodynamic Type
    		234(8)
    9.3 Boundary-Value Problems for the Fokker---Planck Equation (Hopping Phenomenon)
    		242(3)
    9.4 Method of Fast Oscillation Averaging
    		245(2)
    Problems
    		247(6)
    10 Some Other Approximate Approaches to the Problems of Statistical Hydrodynamics
    		253(16)
    10.1 Quasi-Elastic Properties of Isotropic and Stationary Noncompressible Turbulent Media
    		254(4)
    10.2 Sound Radiation by Vortex Motions
    		258(11)
    10.2.1 Sound Radiation by Vortex Lines
    		260(3)
    10.2.2 Sound Radiation by Vortex Rings
    		263(6)
    Part III Examples of Coherent Phenomena in Stochastic Dynamic Systems
    		269(124)
    11 Passive Tracer Clustering and Diffusion in Random Hydrodynamic and Magnetohydrodynamic Flows
    		271(54)
    11.1 General Remarks
    		271(5)
    11.2 Particle Diffusion in Random Velocity Field
    		276(8)
    11.2.1 One-Point Statistical Characteristics
    		276(5)
    11.2.2 Two-Point Statistical Characteristics
    		281(3)
    11.3 Probabilistic Description of Density Field in Random Velocity Field
    		284(7)
    11.4 Probabilistic Description of Magnetic Field and Magnetic Energy in Random Velocity Field
    		291(7)
    11.5 Intergral One-Point Statistical Characteristics of Passive Vector Fields
    		298(21)
    11.5.1 Spatial Correlation Function of Density Field
    		299(3)
    11.5.2 Spatial Correlation Tensor of Density Field Gradient and Dissipation of Density Field Variance
    		302(8)
    11.5.3 Spatial Correlation Function of Magnetic Field
    		310(3)
    11.5.4 On the Magnetic Field Helicity
    		313(2)
    11.5.5 On the Magnetic Field Dissipation
    		315(4)
    Problems
    		319(6)
    12 Wave Localization in Randomly Layered Media
    		325(30)
    12.1 General Remarks
    		325(5)
    12.1.1 Wave Incidence on an Inhomogeneous Layer
    		325(2)
    12.1.2 Source Inside an Inhomogeneous Layer
    		327(3)
    12.2 Statistics of Scattered Field at Layer Boundaries
    		330(9)
    12.2.1 Reflection and Transmission Coefficients
    		330(7)
    12.2.2 Source Inside the Layer of a Medium
    		337(1)
    12.2.3 Statistical Energy Localization
    		338(1)
    12.3 Statistical Theory of Radiative Transfer
    		339(11)
    12.3.1 Normal Wave Incidence on the Layer of Random Media
    		340(7)
    12.3.2 Plane Wave Source Located in Random Medium
    		347(3)
    12.4 Numerical Simulation
    		350(2)
    Problems
    		352(3)
    13 Caustic Structure of Wavefield in Random Media
    		355(38)
    13.1 Input Stochastic Equations and Their Implications
    		355(6)
    13.2 Wavefield Amplitude-Phase Fluctuations. Rytov's Smooth Perturbation Method
    		361(6)
    13.2.1 Random Phase Screen (Δx < <x)
    		365(1)
    13.2.2 Continuous Medium (Δx = x)
    		366(1)
    13.3 Method of Path Integral
    		367(14)
    13.3.1 Asymptotic Analysis of Plane Wave Intensity Fluctuations
    		371(10)
    13.4 Elements of Statistical Topography of Random Intensity Field
    		381(7)
    13.4.1 Weak Intensity Fluctuations
    		383(3)
    13.4.2 Strong Intensity Fluctuations
    		386(2)
    Problems
    		388(5)
    References 393
    Born in 1940 in Moscow, USSR, Valery I. Klyatskin received his secondary education at school in Tbilisi, Georgia, finishing in 1957. Seven years later he graduated from Moscow Institute of Physics and Technology (FIZTEX), whereupon he took up postgraduate studies at the Institute of Atmospheric Physics USSR Academy of Sciences, Moscow gaining the degree of Candidate of Physical and Mathematical Sciences (Ph.D) in 1968. He then continued at the Institute as a researcher, until 1978, when he was appointed as Head of the Wave Process Department at the Pacific Oceanological Institute of the USSR Academy of Sciences, based in Vladivostok. In 1992 Valery I. Klyatskin returned to Institute of Atmospheric Physics Russian Academy of Sciences, Moscow when he was appointed to his present position as Chief Scientist. At the same time he is Chief Scientific Consultant of Pacific Oceanological Institute Russian Academy of Sciences, Vladivostok. In 1977 he obtained a doctorate in Physical and Mathematical Sciences and in 1988 became Research Professor of Theoretical and Mathematical Physics, Russian Academy of Science.
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