Physics of Stochastic Processes : How Randomness Acts in Time

• Autore: Mahnke Reinhard
• Pubblicazione: Weinheim : , : John Wiley & Sons, Incorporated, , 2009; ©2009
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (450 pages)
• Disciplina: 519.23
• Soggetto topico: Stochastic processes; Random measures; Statistical physics; Stochastic processes -- Problems, exercises, etc; Random measures -- Problems, exercises, etc; Statistical physics -- Problems, exercises, etc
• Soggetto genere / forma: Electronic books.
• Altri autori: KaupuzsJevgenijs LubashevskyIhor
• Nota di contenuto: Physics of Stochastic Processes -- Contents -- Preface -- Part I Basic Mathematical Description -- 1 Fundamental Concepts -- 1.1 Wiener Process, Adapted Processes and Quadratic Variation -- 1.2 The Space of Square Integrable Random Variables -- 1.3 The Ito Integral and the Ito Formula -- 1.4 The Kolmogorov Differential Equation and the Fokker-Planck Equation -- 1.5 Special Diffusion Processes -- 1.6 Exercises -- 2 Multidimensional Approach -- 2.1 Bounded Multidimensional Region -- 2.2 From Chapman-Kolmogorov Equation to Fokker-Planck Description -- 2.2.1 The Backward Fokker-Planck Equation -- 2.2.2 Boundary Singularities -- 2.2.3 The Forward Fokker-Planck Equation -- 2.2.4 Boundary Relations -- 2.3 Different Types of Boundaries -- 2.4 Equivalent Lattice Representation of Random Walks Near the Boundary -- 2.4.1 Diffusion Tensor Representations -- 2.4.2 Equivalent Lattice Random Walks -- 2.4.3 Properties of the Boundary Layer -- 2.5 Expression for Boundary Singularities -- 2.6 Derivation of Singular Boundary Scaling Properties -- 2.6.1 Moments of the Walker Distribution and the Generating Function -- 2.6.2 Master Equation for Lattice Random Walks and its General Solution -- 2.6.3 Limit of Multiple-Step Random Walks on Small Time Scales -- 2.6.4 Continuum Limit and a Boundary Model -- 2.7 Boundary Condition for the Backward Fokker-Planck Equation -- 2.8 Boundary Condition for the Forward Fokker-Planck Equation -- 2.9 Concluding Remarks -- 2.10 Exercises -- Part II Physics of Stochastic Processes -- 3 The Master Equation -- 3.1 Markovian Stochastic Processes -- 3.2 The Master Equation -- 3.3 One-Step Processes in Finite Systems -- 3.4 The First-Passage Time Problem -- 3.5 The Poisson Process in Closed and Open Systems -- 3.6 The Two-Level System -- 3.7 The Three-Level System -- 3.8 Exercises -- 4 The Fokker-Planck Equation. ; 4.1 General Fokker-Planck Equations -- 4.2 Bounded Drift-Diffusion in One Dimension -- 4.3 The Escape Problem and its Solution -- 4.4 Derivation of the Fokker-Planck Equation -- 4.5 Fokker-Planck Dynamics in Finite State Space -- 4.6 Fokker-Planck Dynamics with Coordinate-Dependent Diffusion Coefficient -- 4.7 Alternative Method of Solving the Fokker-Planck Equation -- 4.8 Exercises -- 5 The Langevin Equation -- 5.1 A System of Many Brownian Particles -- 5.2 A Traditional View of the Langevin Equation -- 5.3 Additive White Noise -- 5.4 Spectral Analysis -- 5.5 Brownian Motion in Three-Dimensional Velocity Space -- 5.6 Stochastic Differential Equations -- 5.7 The Standard Wiener Process -- 5.8 Arithmetic Brownian Motion -- 5.9 Geometric Brownian Motion -- 5.10 Exercises -- Part III Applications -- 6 One-Dimensional Diffusion -- 6.1 Random Walk on a Line and Diffusion: Main Results -- 6.2 A Drunken Sailor as Random Walker -- 6.3 Diffusion with Natural Boundaries -- 6.4 Diffusion in a Finite Interval with Mixed Boundaries -- 6.5 The Mirror Method and Time Lag -- 6.6 Maximum Value Distribution -- 6.7 Summary of Results for Diffusion in a Finite Interval -- 6.7.1 Reflected Diffusion -- 6.7.2 Diffusion in a Semi-Open System -- 6.7.3 Diffusion in an Open System -- 6.8 Exercises -- 7 Bounded Drift-Diffusion Motion -- 7.1 Drift-Diffusion Equation with Natural Boundaries -- 7.2 Drift-Diffusion Problem with Absorbing and Reflecting Boundaries -- 7.3 Dimensionless Drift-Diffusion Equation -- 7.4 Solution in Terms of Orthogonal Eigenfunctions -- 7.5 First-Passage Time Probability Density -- 7.6 Cumulative Breakdown Probability -- 7.7 The Limiting Case for Large Positive Values of the Control Parameter -- 7.8 A Brief Survey of the Exact Solution -- 7.8.1 Probability Density -- 7.8.2 Outflow Probability Density. ; 7.8.3 First Moment of the Outflow Probability Density -- 7.8.4 Second Moment of the Outflow Probability Density -- 7.8.5 Outflow Probability -- 7.9 Relationship to the Sturm-Liouville Theory -- 7.10 Alternative Method by the Backward Fokker-Planck Equation -- 7.11 Roots of the Transcendental Equation -- 7.12 Exercises -- 8 The Ornstein-Uhlenbeck Process -- 8.1 Definitions and Properties -- 8.2 The Ornstein-Uhlenbeck Process and its Solution -- 8.3 The Ornstein-Uhlenbeck Process with Linear Potential -- 8.4 The Exponential Ornstein-Uhlenbeck Process -- 8.5 Outlook on Econophysics -- 8.6 Exercises -- 9 Nucleation in Supersaturated Vapors -- 9.1 Dynamics of First-Order Phase Transitions in Finite Systems -- 9.2 Condensation of Supersaturated Vapor -- 9.3 The General Multi-Droplet Scenario -- 9.4 Detailed Balance and Free Energy -- 9.5 Relaxation to the Free Energy Minimum -- 9.6 Chemical Potentials -- 9.7 Exercises -- 10 Vehicular Traffic -- 10.1 The Car-Following Theory -- 10.2 The Optimal Velocity Model and its Langevin Approach -- 10.3 Traffic Jam Formation on a Circular Road -- 10.4 Metastability Near Phase Transitions in Traffic Flow -- 10.5 Car Cluster Formation as First-Order Phase Transition -- 10.6 Thermodynamics of Traffic Flow -- 10.7 Exercises -- 11 Noise-Induced Phase Transitions -- 11.1 Equilibrium and Nonequilibrium Phase Transitions -- 11.2 Types of Stochastic Differential Equations -- 11.3 Transformation of Random Variables -- 11.4 Forms of the Fokker-Planck Equation -- 11.5 The Verhulst Model of Third Order -- 11.6 The Genetic Model -- 11.7 Noise-Induced Instability in Geometric Brownian Motion -- 11.8 System Dynamics with Stagnation -- 11.9 Oscillator with Dynamical Traps -- 11.10 Dynamics with Traps in a Chain of Oscillators -- 11.11 Self-Freezing Model for Multi-Lane Traffic -- 11.12 Exercises -- 12 Many-Particle Systems. ; 12.1 Hopping Models with Zero-Range Interaction -- 12.2 The Zero-Range Model of Traffic Flow -- 12.3 Transition Rates and Phase Separation -- 12.4 Metastability -- 12.5 Monte Carlo Simulations of the Hopping Model -- 12.6 Fundamental Diagram of the Zero-Range Model -- 12.7 Polarization Kinetics in Ferroelectrics with Fluctuations -- 12.8 Exercises -- Epilog -- References -- Index.
• Sommario/riassunto: Based on lectures given by one of the authors with many years of experience in teaching stochastic processes, this textbook is unique in combining basic mathematical and physical theory with numerous simple and sophisticated examples as well as detailed calculations. In addition, applications from different fields are included so as to strengthen the background learned in the first part of the book. With its exercises at the end of each chapter (and solutions only available to lecturers) this book will benefit students and researchers at different educational levels. Solutions manual available for lecturers on www.wiley-vch.de.
• Titolo autorizzato: Physics of Stochastic Processes
• ISBN: 9783527626106; 9783527408405
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910809145603321
• Lo trovi qui: Univ. Federico II