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# Stochastic Equations through the Eye of the Physicist,

**Valery Klyatskin**

ISBN **0444517979**

Pages **556**

Description

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.

Contents

Contents Preface Introduction I Dynamical description of stochastic systems 1 Examples, basic problems, peculiar features of solutions 1.1 Ordinary differential equations: initial value problems 1.1.1 Particle under the random velocity field 1.1.2 Particles under the random velocity field 1.1.3 Particles under random forces 1.1.4 Systems with the blow-up singularities 1.1.5 Oscillator with randomly varying frequency (stochastic parametric resonance) 1.2 Linear ordinary differential equations: boundary-value problems 1.2.1 Plane waves in layered media: a wave incident on a medium layer 1.2.2 Plane waves in layered media: the source inside the medium 1.2.3 Plane waves in layered media: the two-layer model 1.3 First-order partial differential equations 1.3.1 Linear first-order partial differential equations: passive tracer in random velocity field 1.3.2 Quasilinear equations 1.3.3 Boundary-value problems for nonlinear ordinary differential equations 1.3.4 Nonlinear first-order partial differential equations 1.4 Partial differential equations of higher orders 1.4.1 Stationary problems for Maxwell’s equations 1.4.2 The Helmholtz equation (boundary-value problem) and the parabolic equation of quasioptics (waves in randomly inhomogeneous media) 1.4.3 The Navier–Stokes equation: random forces in hydrodynamic theory of turbulence 1.4.4 Equations of geophysical hydrodynamics 1.5 Solution dependence on medium parameters and initial value 1.5.1 Principle of dynamic causality 1.5.2 Solution dependence on initial value 2 Indicator function and Liouville equation 2.1 Ordinary differential equations 2.2 First-order partial differential equations 2.2.1 Linear equations 2.2.2 Quasilinear equations 2.2.3 General-form nonlinear equations 2.3 Higher-order partial differential equations 2.3.1 Parabolic equation of quasioptics 2.3.2 Random forces in hydrodynamic theory of turbulence II Stochastic equations 3 Random quantities, processes and fields 3.1 Random quantities and their characteristics 3.2 Random processes, fields, and their characteristics 3.2.1 General remarks 3.2.2 Statistical topography of random processes and fields 3.2.3 Gaussian random process 3.2.4 Discontinuous random processes 3.3 Markovian processes 3.3.1 General properties 3.3.2 Characteristic functional of the Markovian process 4 Correlation splitting 4.1 General remarks 4.2 Gaussian process 4.3 Poisson process 4.4 Telegrapher’s random process 4.5 Generalized telegrapher’s random process 4.6 General-form Markovian processes 4.7 Delta-correlated random processes 4.7.1 Asymptotic meaning of delta-correlated processes and fields 5 General approaches to analyzing stochastic dynamic systems 5.1 Ordinary differential equations 5.2 Partial differential equations 5.2.1 Passive tracer transfer in random field of velocities 5.2.2 Parabolic equation of quasi-optics 5.2.3 Random forces in the theory of hydrodynamic turbulence 5.3 Stochastic integral equations (methods of quantum field theory in the dynamics of stochastic systems) 5.3.1 Linear integral equations 5.3.2 Nonlinear integral equations 5.4 Completely solvable stochastic dynamic systems 5.4.1 Ordinary differential equations 5.4.2 Partial differential equations 5.5 Delta-correlated fields and processes 5.5.1 One-dimensional nonlinear differential equation 5.5.2 Linear operator equation 5.5.3 Partial differential equations 6 Stochastic equations with the Markovian fluctuations of parameters 6.1 Telegrapher’s processes 6.1.1 System of linear operator equations 6.1.2 One-dimension nonlinear differential equation 6.1.3 Particle in the one-dimension potential field 6.1.4 Ordinary differential equation of the n-th order 6.1.5 Statistical interpretation of telegrapher’s equation 6.2 Generalized telegrapher’s process 6.2.1 Stochastic linear equation 6.2.2 One-dimensional nonlinear differential equation 6.2.3 Ordinal differential equation of the n-th order 6.3 Gaussian Markovian processes 6.3.1 Stochastic linear equation 6.3.2 Ordinal differential equation of the n-th order 6.3.3 The square of the Gaussian Markovian process 6.4 Markovian processes with finite-dimensional phase space 6.4.1 Two-state process 6.5 Causal stochastic integral equations 6.5.1 Telegrapher’s random process 6.5.2 Generalized telegrapher’s random process 6.5.3 Gaussian Markovian process III Asymptotic and approximate methods for analyzing stochastic equations 7 Gaussian random field delta-correlated in time (ordinary differential equations) 7.1 The Fokker–Planck equation 7.2 Transitional probability distributions 7.3 Applicability range of the Fokker–Planck equation 7.3.1 Langevin equation 8 Methods for solving and analyzing the Fokker-Planck equation 8.1 System of linear equations 8.1.1 Wiener random process 8.1.2 Logarithmic-normal random process 8.2 Integral transformations 8.3 Steady-state solutions of the Fokker–Planck equation 8.3.1 One-dimensional nonlinear differential equation 8.3.2 Hamiltonian systems 8.3.3 Systems of hydrodynamic type 8.4 Boundary-value problems for the Fokker-Planck equation (transfer phenomena) 8.4.1 Transfer phenomena in regular systems 8.4.2 Transfer phenomena in singular systems 8.5 Asymptotic and approximate methods of solving the Fokker-Plank equation 8.5.1 Asymptotic expansion 8.5.2 Method of cumulant expansions 8.5.3 Method of fast oscillation averaging 9 Gaussian delta-correlated random field (causal integral equations) 9.1 Causal integral equation 9.2 Statistical averaging 10 Diffusion approximation IV Coherent phenomena in stochastic dynamic systems 11 Passive tracer clustering and diffusion in random hydrodynamic flows 11.1 General remarks 11.2 Statistical description 11.2.1 Lagrangian description (particle diffusion) 11.2.2 Eulerian description 11.3 Additional factors 11.3.1 Plane-parallel mean shear 11.3.2 Effect of molecular diffusion 11.3.3 Consideration of finite temporal correlation radius 12 Wave localization in randomly layered media 12.1 Statistics of scattered field at layer boundaries 12.1.1 Reflection and transmission coefficients 12.1.2 Source inside the layer of a medium 12.1.3 Statistical energy localization 12.1.4 Diffusion approximation 12.2 Statistical description of a wave field in random medium 12.2.1 Normal wave incidence on the layer of random media 12.2.2 Plane wave source located in random medium 12.2.3 Numerical simulation 12.3 Eigenvalue and eigenfunction statistics 12.3.1 General remarks 12.3.2 Statistical averaging 12.4 Multidimensional wave problems in layered random media 12.4.1 Nonstationary problems 12.4.2 Point source in randomly layered medium 12.5 Two-layer model of the medium 12.5.1 Formulation of boundary-value problems 12.5.2 Statistical description 13 Wave propagation in random inhomogeneous medium 13.1 Method of stochastic equation 13.1.1 Stochastic equations and their implication 13.1.2 Delta-correlated approximation of the media parameters 13.1.3 Conditions for the applicability of the delta-correlation approximation of the medium parameters fluctuations, and diffusion approximation for the wave field 13.1.4 Wavefield amplitude–phase fluctuations (the Smooth Perturbation Method) 13.2 Geometrical optics approximation in randomly inhomogeneous media 13.2.1 Ray diffusion in random media (Lagrangian description) 13.2.2 Caustics formation in randomly inhomogeneous media 13.2.3 Wavefield amplitude–phase fluctuations (Eulerian description) 13.3 Method of path integral 13.3.1 Statistical description of wavefield 13.3.2 Asymptotical analysis of intensity fluctuations of plane wave 13.3.3 Caustical structure of wavefield in random media 14 Some problems of statistical hydrodynamics 14.1 Quasicompressible properties of isotropic and stationar noncompressible turbulent media 14.2 Radiation of sound by vortex systems 14.2.1 Sound radiation by vortex lines 14.2.2 Sound radiation by vortex rings V Appendix A Variation (functional) derivatives B Fundamental solutions of wave problems in empty and layered media B.1 The case of empty space B.2 The case of layered space C Imbedding method in boundary-value wave problems C.1 Boundary-value problems for ordinary differential equations C.2 Stationary boundary-value wave problems C.2.1 One-dimensional stationary boundary-value wave problems C.2.2 Waves in periodically inhomogeneous media C.2.3 Boundary-value stationary nonlinear wave problem of self-action C.2.4 Stationary multidimensional boundary-value problem C.3 One-dimensional nonstationary boundary-value wave problem C.3.1 Nonsteady medium C.3.2 Steady medium C.3.3 One-dimensional nonlinear wave problem Bibliography Index