High-dimensional Nonlinear Diffusion Stochastic Processes: Modelling for Engineering ApplicationsWorld Scientific, 2001 - 297 من الصفحات 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. |
المحتوى
Chapter 1 Introductory Chapter | 1 |
Chapter 2 Diffusion Processes | 63 |
Chapter 3 Invariant Diffusion Processes | 85 |
Chapter 4 Stationary Diffusion Processes | 107 |
Chapter 5 Itos Stochastic Partial Differential Equations as NonMarkov Models Leading to HighDimensional Diffusion Processes ... | 141 |
Chapter 6 Itos Stochastic Partial Differential Equations for Electron Fluids in Semiconductors ... | 163 |
Chapter 7 Distinguishing Features of Engineering Applications | 197 |
Chapter 8 AnalyticalNumerical Approach to Engineering Problems and Common Analytical Techniques ... | 201 |
Nonlinear Friction and Unbounded Stationary Probability Density of the Particle Velocity in Uniform Fluid ... | 213 |
Appendix D Proofs of the Theorems in Chapter 2 and Other Details | 231 |
Appendix E Proofs of the Theorems in Chapter 4 | 243 |
Appendix F Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and ChemicalReaction GenerationRecom... | 247 |
Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in ThreeDimensional Space ... | 255 |
Appendix H Resources for Engineering Parallel Computing under Windows 95 | 261 |
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281 | |
Solutions of the Cauchy Problems for Ordinary Differential Equation System ... | 205 |
Appendix B SignaltoNoise Ratio | 209 |
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عبارات ومصطلحات مألوفة
analytical Appendix approximation Arnold assertion assumption asymptotic Banach spaces basis Bellomo Chapter computational considered corresponding covariance Definition Demir depend derivatives described determined deterministic diffusion functions discussed drift and diffusion electron elementary event engineering entries evaluate example expression feature fixed zEQ fluid formula functions g holds initial condition initial-value problem instance integral ISODE ISODE system ISPDE ISPIDE Itô's stochastic linear macroscopic Mamontov and Willander Markov process mathematical means method momentum-relaxation noise nonlinear Note ODE system parameter particle Phys physical present book probability density quantity random variable Rd Rd Rd+1 Remark respect right-hand side scalar Section semiconductor SF-ISPDE simulation Soize solution spectral density Stochastic Differential Equations stochastic process stochastic resonance techniques Theorem 2.1 theory tion transition probability density treatment uniformly valid variance matrix velocity well-known Wiener process