Mamontov, Yevgeny.
High-Dimensional Nonlinear Diffusion Stochastic Processes : Modelling for Engineering Applications. - 1 online resource (322 pages) - Series on Advances in Mathematics for Applied Sciences Ser. ; v.56 . - Series on Advances in Mathematics for Applied Sciences Ser. .
Intro -- Contents -- Preface -- Chapter 1 Introductory Chapter -- 1.1 Prerequisites for Reading -- 1.2 Random Variable. Stochastic Process. Random Field. High-Dimensional Process. One-Point Process -- 1.3 Two-Point Process. Expectation. Markov Process. Example of Non-Markov Process Associated with Multidimensional Markov Process -- 1.4 Preceding Subsequent and Transition Probability Densities. The Chapman-Kolmogorov Equation. Initial Condition for Markov Process -- 1.4.1 The Chapman-Kolmogorov equation -- 1.4.2 Initial condition for Markov process -- 1.5 Homogeneous Markov Process. Example of Markov Process: The Wiener Process -- 1.6 Expectation Variance and Standard Deviations of Markov Process -- 1.7 Invariant and Stationary Markov Processes. Covariance. Spectral Densities -- 1.8 Diffusion Process -- 1.9 Example of Diffusion Processes: Solutions of Ito's Stochastic Ordinary Differential Equation -- 1.10 The Kolmogorov Backward Equation -- 1.11 Figures of Merit. Diffusion Modelling of High-Dimensional Systems -- 1.12 Common Analytical Techniques to Determine Probability Densities of Diffusion Processes. The Kolmogorov Forward Equation -- 1.12.1 Probability density -- 1.12.2 Invariant probability density -- 1.12.3 Stationary probability density -- 1.13 The Purpose and Content of This Book -- Chapter 2 Diffusion Processes -- 2.1 Introduction -- 2.2 Time-Derivatives of Expectation and Variance -- 2.3 Ordinary Differential Equation Systems for Expectation -- 2.3.1 The first-order system -- 2.3.2 The second-order system -- 2.3.3 Systems of the higher orders -- 2.4 Models for Noise-Induced Phenomena in Expectation -- 2.4.1 The case of stochastic resonance -- 2.4.2 Practically efficient implementation of the second-order system -- 2.5 Ordinary Differential Equation System for Variance -- 2.5.1 Damping matrix. 2.5.2 The uncorrelated-matrixes approximation -- 2.5.3 Nonlinearity of the drift function -- 2.5.4 Fundamental limitation of the state-space-independent approximations for the diffusion and damping matrixes -- 2.6 The Steady-State Approximation for The Probability Density -- Chapter 3 Invariant Diffusion Processes -- 3.1 Introduction -- 3.2 Preliminary Remarks -- 3.3 Expectation. The Finite-Equation Method -- 3.4 Explicit Expression for Variance -- 3.5 The Simplified Detailed-Balance Approximation for Invariant Probability Density -- 3.5.1 Partial differential equation for logarithm of the density -- 3.5.2 Truncated equation for the logarithm and the detailed-balance equation -- 3.5.3 Case of the detailed balance -- 3.5.4 The detailed-balance approximation -- 3.5.5 The simplified detailed-balance approximation. Theorem on the approximating density -- 3.6 Analytical-Numerical Approach to Non-Invariant and Invariant Diffusion Processes -- 3.6.1 Choice of the bounded domain of the integration -- 3.6.2 Evaluation of the multifold integrals. The Monte Carlo technique -- 3.6.3 Summary of the approach -- 3.7 Discussion -- Chapter 4 Stationary Diffusion Processes -- 4.1 Introduction -- 4.2 Previous Results Related to Covariance and Spectral-Density Matrixes -- 4.3 Time-Separation Derivative of Covariance in the Limit Case of Zero Time Separation -- 4.4 Flicker Effect -- 4.5 Time-Separation Derivative of Covariance in the General Case -- 4.6 Case of the Uncorrelated Matrixes -- 4.7 Representations for Spectral Density in the Uncorrelated-Matrixes Case -- 4.8 Example: Comparison of the Dampings for a Particle Near the Minimum of Its Potential Energy -- 4.9 The Deterministic-Transition Approximation -- 4.10 Example: Non-exponential Covariance of Velocity of a Particle in a Fluid -- 4.10.1 Covariance in the general case. 4.10.2 Covariance and the qualities related to it in a simple fluid -- 4.10.3 Case of the hard-sphere fluid -- 4.10.4 Summary of the procedure in the general case -- 4.11 Analytical-Numerical Approach to Stationary Process -- 4.11.1 Practical issues -- 4.11.2 Summary of the approach -- 4.12 Discussion -- Chapter 5 Ito's Stochastic Partial Differential Equations as Non-Markov Models Leading to High-Dimensional Diffusion Processes -- 5.1 Introduction -- 5.2 Various Types of Ito's Stochastic Differential Equations -- 5.3 Method to Reduce ISPIDE to System of ISODEs -- 5.3.1 Projection approach -- 5.3.2 Stochastic collocation method -- 5.3.3 Stochastic-adaptive-interpolation method -- 5.4 Related Computational Issues -- 5.5 Discussion -- Chapter 6 Ito's Stochastic Partial Differential Equations for Electron Fluids in Semiconductors -- 6.1 Introduction -- 6.2 Microscopic Phenomena in Macroscopic Models of Multiparticle Systems -- 6.2.1 Microscopic random walks in deterministic macroscopic models of multiparticle systems -- 6.2.2 Macroscale mesoscale and microscale domains in terms of the wave-diffusion equation -- 6.2.3 Stochastic generalization of the deterministic macroscopic models of multiparticle systems -- 6.3 The ISPDE System for Electron Fluid in n-Type Semiconductor -- 6.3.1 Deterministic model for electron fluid in semiconductor -- 6.3.2 Mesoscopic wave-diffusion equations in the deterministic fluid-dynamic model -- 6.3.3 Stochastic generalization of the deterministic fluid-dynamic model. The semiconductor-fluid ISPDE system -- 6.4 Semiconductor Noises and the SF-ISPDE System: Discussion and Future Development -- 6.4.1 The SF-ISPDE system in connection with semiconductor noises -- 6.4.2 Some directions for future development -- Chapter 7 Distinguishing Features of Engineering Applications -- 7.1 High-Dimensional Diffusion Processes. 7.2 Efficient Multiple Analysis -- 7.3 Reasonable Amount of the Main Computer Memory -- 7.4 Real-Time Signal Transformation by Diffusion Process -- Chapter 8 Analytical-Numerical Approach to Engineering Problems and Common Analytical Techniques -- 8.1 Analytical-Numerical Approach to Engineering Problems -- 8.2 Severe Practical Limitations of Common Analytical Techniques in the High-Dimensional Case. Possible Alternatives -- Appendix A Example of Markov Processes: Solutions of the Cauchy Problems for Ordinary Differential Equation System -- Appendix B Signal-to-Noise Ratio -- Appendix C Example of Application of Corollary 1.2: Nonlinear Friction and Unbounded Stationary Probability Density of the Particle Velocity in Uniform Fluid -- C.1 Description of the Model -- C.2 Energy-Independent Momentum-Relaxation Time. Equilibrium Probability Density -- C.3 Energy-Dependent Momentum-Relaxation Time: General Case -- C.4 Energy-Dependent Momentum-Relaxation Time: Case of Simple Fluid -- Appendix D Proofs of the Theorems in Chapter 2 and Other Details -- D.1 Proof of Theorem 2.1 -- D.2 Proof of Theorem 2.2 -- D.3 Green's Formula for the Differential Operator of Kolmogorov's Backward Equation -- D.4 Proof of Theorem 2.3 -- D.5 Quasi-Neutral Equilibrium Point -- D.6 Proof of Theorem 2.4 -- Appendix E Proofs of the Theorems in Chapter 4 -- E.1 Proof of Lemma 4.1 -- E.2 Proof of Theorem 4.2 -- E.3 Proof of Theorem 4.3 -- E.4 Proof of Theorem 4.4 -- Appendix F Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and Chemical-Reaction / Generation-Recombination Noise -- Appendix G Example: Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in Three-Dimensional Space -- Appendix H Resources for Engineering Parallel Computing under Windows 95 -- Bibliography -- Index.
This book is the first one devoted to high-dimensional (or large-scale) diffusion stochastic processes (DSPs) with nonlinear coefficients. These processes are closely associated with nonlinear Ito's stochastic ordinary differential equations (ISODEs) and with the space-discretized versions of nonlinear Ito's stochastic partial integro-differential equations. The latter models include Ito's stochastic partial differential equations (ISPDEs). The book presents the new analytical treatment which can serve as the basis of a combined, analytical-numerical approach to greater computational efficiency in engineering problems. A few examples discussed in the book include: the high-dimensional DSPs described with the ISODE systems for semiconductor circuits; the nonrandom model for stochastic resonance (and other noise-induced phenomena) in high-dimensional DSPs; the modification of the well-known stochastic-adaptive-interpolation method by means of bases of function spaces; ISPDEs as the tool to consistently model non-Markov phenomena; the ISPDE system for semiconductor devices; the corresponding classification of charge transport in macroscale, mesoscale and microscale semiconductor regions based on the wave-diffusion equation; the fully time-domain nonlinear-friction aware analytical model for the velocity covariance of particle of uniform fluid, simple or dispersed; the specific time-domain analytics for the long, non-exponential "tails" of the velocity in case of the hard-sphere fluid. These examples demonstrate not only the capabilities of the developed techniques but also emphasize the usefulness of the complex-system-related approaches to solve some problems which have not been solved with the traditional, statistical-physics methods yet. From this veiwpoint, the book can be regarded as a kind of complement to such books as "Introduction to the Physics of Complex Systems. The Mesoscopic Approach to Fluctuations, Nonlinearity and Self-Organization" by Serra, Andretta, Compiani and Zanarini, "Stochastic Dynamical Systems. Concepts, Numerical Methods, Data Analysis" and "Statistical Physics: An Advanced Approach with Applications" by Honerkamp which deal with physics of complex systems, some of the corresponding analysis methods and an innovative, stochastics-based vision of theoretical physics. To facilitate the reading by nonmathematicians, the introductory chapter outlines the basic notions and results of theory of Markov and diffusion stochastic processes without involving the measure-theoretical approach. This presentation is based on probability densities commonly used in engineering and applied sciences. Contents: Introductory Chapter; Diffusion Processes; Invariant Diffusion Processes; Stationary Diffusion Processes; Itô's Stochastic Partial Differential Equations as Non-Markov Models Leading to High-Dimensional Diffusion Processes; Itô's Stochastic Partial Differential Equations for Electron Fluids in Semiconductors; Distinguishing Features of Engineering Applications; Analytical-Numerical Approach to Engineering Problems and Common Analytical Techniques; Appendices: Example of Markov Processes: Solutions of the Cauchy Problems for Ordinary Differential Equation System; Signal-to-Noise Ratio; Example of Application of Corollary 1.2: Nonlinear Friction and Unbounded Stationary Probability Density of the Particle Velocity in Uniform Fluid; Proofs of the Theorems in Chapter 2 and Other Details; Proofs of the Theorems in Chapter 4; Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and Chemical-Reaction/Generation-Recombination Noise; Example: Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in Three-Dimensional Space; Resources for Engineering Parallel Computing under Windows 95. Readership: Nonmathematicians (e.g., theoretical physicists, engineers in industry, specialists in models for finance or biology, computing scientists), mathematicians, undergraduate and postgraduate students of the corresponding specialties, managers in applied sciences and engineering dealing with the advancements in the related fields, any specialists who use diffusion stochastic processes to model high-dimensional (or large-scale) nonlinear stochastic systems.
9789812810540
Differential equations, Nonlinear.
Diffusion processes.
Engineering -- Mathematical models.
Stochastic processes.
Electronic books.
TA342.M35 2001
620
High-Dimensional Nonlinear Diffusion Stochastic Processes : Modelling for Engineering Applications. - 1 online resource (322 pages) - Series on Advances in Mathematics for Applied Sciences Ser. ; v.56 . - Series on Advances in Mathematics for Applied Sciences Ser. .
Intro -- Contents -- Preface -- Chapter 1 Introductory Chapter -- 1.1 Prerequisites for Reading -- 1.2 Random Variable. Stochastic Process. Random Field. High-Dimensional Process. One-Point Process -- 1.3 Two-Point Process. Expectation. Markov Process. Example of Non-Markov Process Associated with Multidimensional Markov Process -- 1.4 Preceding Subsequent and Transition Probability Densities. The Chapman-Kolmogorov Equation. Initial Condition for Markov Process -- 1.4.1 The Chapman-Kolmogorov equation -- 1.4.2 Initial condition for Markov process -- 1.5 Homogeneous Markov Process. Example of Markov Process: The Wiener Process -- 1.6 Expectation Variance and Standard Deviations of Markov Process -- 1.7 Invariant and Stationary Markov Processes. Covariance. Spectral Densities -- 1.8 Diffusion Process -- 1.9 Example of Diffusion Processes: Solutions of Ito's Stochastic Ordinary Differential Equation -- 1.10 The Kolmogorov Backward Equation -- 1.11 Figures of Merit. Diffusion Modelling of High-Dimensional Systems -- 1.12 Common Analytical Techniques to Determine Probability Densities of Diffusion Processes. The Kolmogorov Forward Equation -- 1.12.1 Probability density -- 1.12.2 Invariant probability density -- 1.12.3 Stationary probability density -- 1.13 The Purpose and Content of This Book -- Chapter 2 Diffusion Processes -- 2.1 Introduction -- 2.2 Time-Derivatives of Expectation and Variance -- 2.3 Ordinary Differential Equation Systems for Expectation -- 2.3.1 The first-order system -- 2.3.2 The second-order system -- 2.3.3 Systems of the higher orders -- 2.4 Models for Noise-Induced Phenomena in Expectation -- 2.4.1 The case of stochastic resonance -- 2.4.2 Practically efficient implementation of the second-order system -- 2.5 Ordinary Differential Equation System for Variance -- 2.5.1 Damping matrix. 2.5.2 The uncorrelated-matrixes approximation -- 2.5.3 Nonlinearity of the drift function -- 2.5.4 Fundamental limitation of the state-space-independent approximations for the diffusion and damping matrixes -- 2.6 The Steady-State Approximation for The Probability Density -- Chapter 3 Invariant Diffusion Processes -- 3.1 Introduction -- 3.2 Preliminary Remarks -- 3.3 Expectation. The Finite-Equation Method -- 3.4 Explicit Expression for Variance -- 3.5 The Simplified Detailed-Balance Approximation for Invariant Probability Density -- 3.5.1 Partial differential equation for logarithm of the density -- 3.5.2 Truncated equation for the logarithm and the detailed-balance equation -- 3.5.3 Case of the detailed balance -- 3.5.4 The detailed-balance approximation -- 3.5.5 The simplified detailed-balance approximation. Theorem on the approximating density -- 3.6 Analytical-Numerical Approach to Non-Invariant and Invariant Diffusion Processes -- 3.6.1 Choice of the bounded domain of the integration -- 3.6.2 Evaluation of the multifold integrals. The Monte Carlo technique -- 3.6.3 Summary of the approach -- 3.7 Discussion -- Chapter 4 Stationary Diffusion Processes -- 4.1 Introduction -- 4.2 Previous Results Related to Covariance and Spectral-Density Matrixes -- 4.3 Time-Separation Derivative of Covariance in the Limit Case of Zero Time Separation -- 4.4 Flicker Effect -- 4.5 Time-Separation Derivative of Covariance in the General Case -- 4.6 Case of the Uncorrelated Matrixes -- 4.7 Representations for Spectral Density in the Uncorrelated-Matrixes Case -- 4.8 Example: Comparison of the Dampings for a Particle Near the Minimum of Its Potential Energy -- 4.9 The Deterministic-Transition Approximation -- 4.10 Example: Non-exponential Covariance of Velocity of a Particle in a Fluid -- 4.10.1 Covariance in the general case. 4.10.2 Covariance and the qualities related to it in a simple fluid -- 4.10.3 Case of the hard-sphere fluid -- 4.10.4 Summary of the procedure in the general case -- 4.11 Analytical-Numerical Approach to Stationary Process -- 4.11.1 Practical issues -- 4.11.2 Summary of the approach -- 4.12 Discussion -- Chapter 5 Ito's Stochastic Partial Differential Equations as Non-Markov Models Leading to High-Dimensional Diffusion Processes -- 5.1 Introduction -- 5.2 Various Types of Ito's Stochastic Differential Equations -- 5.3 Method to Reduce ISPIDE to System of ISODEs -- 5.3.1 Projection approach -- 5.3.2 Stochastic collocation method -- 5.3.3 Stochastic-adaptive-interpolation method -- 5.4 Related Computational Issues -- 5.5 Discussion -- Chapter 6 Ito's Stochastic Partial Differential Equations for Electron Fluids in Semiconductors -- 6.1 Introduction -- 6.2 Microscopic Phenomena in Macroscopic Models of Multiparticle Systems -- 6.2.1 Microscopic random walks in deterministic macroscopic models of multiparticle systems -- 6.2.2 Macroscale mesoscale and microscale domains in terms of the wave-diffusion equation -- 6.2.3 Stochastic generalization of the deterministic macroscopic models of multiparticle systems -- 6.3 The ISPDE System for Electron Fluid in n-Type Semiconductor -- 6.3.1 Deterministic model for electron fluid in semiconductor -- 6.3.2 Mesoscopic wave-diffusion equations in the deterministic fluid-dynamic model -- 6.3.3 Stochastic generalization of the deterministic fluid-dynamic model. The semiconductor-fluid ISPDE system -- 6.4 Semiconductor Noises and the SF-ISPDE System: Discussion and Future Development -- 6.4.1 The SF-ISPDE system in connection with semiconductor noises -- 6.4.2 Some directions for future development -- Chapter 7 Distinguishing Features of Engineering Applications -- 7.1 High-Dimensional Diffusion Processes. 7.2 Efficient Multiple Analysis -- 7.3 Reasonable Amount of the Main Computer Memory -- 7.4 Real-Time Signal Transformation by Diffusion Process -- Chapter 8 Analytical-Numerical Approach to Engineering Problems and Common Analytical Techniques -- 8.1 Analytical-Numerical Approach to Engineering Problems -- 8.2 Severe Practical Limitations of Common Analytical Techniques in the High-Dimensional Case. Possible Alternatives -- Appendix A Example of Markov Processes: Solutions of the Cauchy Problems for Ordinary Differential Equation System -- Appendix B Signal-to-Noise Ratio -- Appendix C Example of Application of Corollary 1.2: Nonlinear Friction and Unbounded Stationary Probability Density of the Particle Velocity in Uniform Fluid -- C.1 Description of the Model -- C.2 Energy-Independent Momentum-Relaxation Time. Equilibrium Probability Density -- C.3 Energy-Dependent Momentum-Relaxation Time: General Case -- C.4 Energy-Dependent Momentum-Relaxation Time: Case of Simple Fluid -- Appendix D Proofs of the Theorems in Chapter 2 and Other Details -- D.1 Proof of Theorem 2.1 -- D.2 Proof of Theorem 2.2 -- D.3 Green's Formula for the Differential Operator of Kolmogorov's Backward Equation -- D.4 Proof of Theorem 2.3 -- D.5 Quasi-Neutral Equilibrium Point -- D.6 Proof of Theorem 2.4 -- Appendix E Proofs of the Theorems in Chapter 4 -- E.1 Proof of Lemma 4.1 -- E.2 Proof of Theorem 4.2 -- E.3 Proof of Theorem 4.3 -- E.4 Proof of Theorem 4.4 -- Appendix F Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and Chemical-Reaction / Generation-Recombination Noise -- Appendix G Example: Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in Three-Dimensional Space -- Appendix H Resources for Engineering Parallel Computing under Windows 95 -- Bibliography -- Index.
This book is the first one devoted to high-dimensional (or large-scale) diffusion stochastic processes (DSPs) with nonlinear coefficients. These processes are closely associated with nonlinear Ito's stochastic ordinary differential equations (ISODEs) and with the space-discretized versions of nonlinear Ito's stochastic partial integro-differential equations. The latter models include Ito's stochastic partial differential equations (ISPDEs). The book presents the new analytical treatment which can serve as the basis of a combined, analytical-numerical approach to greater computational efficiency in engineering problems. A few examples discussed in the book include: the high-dimensional DSPs described with the ISODE systems for semiconductor circuits; the nonrandom model for stochastic resonance (and other noise-induced phenomena) in high-dimensional DSPs; the modification of the well-known stochastic-adaptive-interpolation method by means of bases of function spaces; ISPDEs as the tool to consistently model non-Markov phenomena; the ISPDE system for semiconductor devices; the corresponding classification of charge transport in macroscale, mesoscale and microscale semiconductor regions based on the wave-diffusion equation; the fully time-domain nonlinear-friction aware analytical model for the velocity covariance of particle of uniform fluid, simple or dispersed; the specific time-domain analytics for the long, non-exponential "tails" of the velocity in case of the hard-sphere fluid. These examples demonstrate not only the capabilities of the developed techniques but also emphasize the usefulness of the complex-system-related approaches to solve some problems which have not been solved with the traditional, statistical-physics methods yet. From this veiwpoint, the book can be regarded as a kind of complement to such books as "Introduction to the Physics of Complex Systems. The Mesoscopic Approach to Fluctuations, Nonlinearity and Self-Organization" by Serra, Andretta, Compiani and Zanarini, "Stochastic Dynamical Systems. Concepts, Numerical Methods, Data Analysis" and "Statistical Physics: An Advanced Approach with Applications" by Honerkamp which deal with physics of complex systems, some of the corresponding analysis methods and an innovative, stochastics-based vision of theoretical physics. To facilitate the reading by nonmathematicians, the introductory chapter outlines the basic notions and results of theory of Markov and diffusion stochastic processes without involving the measure-theoretical approach. This presentation is based on probability densities commonly used in engineering and applied sciences. Contents: Introductory Chapter; Diffusion Processes; Invariant Diffusion Processes; Stationary Diffusion Processes; Itô's Stochastic Partial Differential Equations as Non-Markov Models Leading to High-Dimensional Diffusion Processes; Itô's Stochastic Partial Differential Equations for Electron Fluids in Semiconductors; Distinguishing Features of Engineering Applications; Analytical-Numerical Approach to Engineering Problems and Common Analytical Techniques; Appendices: Example of Markov Processes: Solutions of the Cauchy Problems for Ordinary Differential Equation System; Signal-to-Noise Ratio; Example of Application of Corollary 1.2: Nonlinear Friction and Unbounded Stationary Probability Density of the Particle Velocity in Uniform Fluid; Proofs of the Theorems in Chapter 2 and Other Details; Proofs of the Theorems in Chapter 4; Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and Chemical-Reaction/Generation-Recombination Noise; Example: Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in Three-Dimensional Space; Resources for Engineering Parallel Computing under Windows 95. Readership: Nonmathematicians (e.g., theoretical physicists, engineers in industry, specialists in models for finance or biology, computing scientists), mathematicians, undergraduate and postgraduate students of the corresponding specialties, managers in applied sciences and engineering dealing with the advancements in the related fields, any specialists who use diffusion stochastic processes to model high-dimensional (or large-scale) nonlinear stochastic systems.
9789812810540
Differential equations, Nonlinear.
Diffusion processes.
Engineering -- Mathematical models.
Stochastic processes.
Electronic books.
TA342.M35 2001
620