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Physics of Stochastic Processes : How Randomness Acts in Time
Physics of Stochastic Processes : How Randomness Acts in Time
Autore Mahnke Reinhard
Edizione [1st ed.]
Pubbl/distr/stampa Weinheim : , : John Wiley & Sons, Incorporated, , 2009
Descrizione fisica 1 online resource (450 pages)
Disciplina 519.23
Altri autori (Persone) KaupuzsJevgenijs
LubashevskyIhor
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.
ISBN 9783527626106
9783527408405
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
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.
Record Nr. UNINA-9910795983603321
Mahnke Reinhard  
Weinheim : , : John Wiley & Sons, Incorporated, , 2009
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Physics of Stochastic Processes : How Randomness Acts in Time
Physics of Stochastic Processes : How Randomness Acts in Time
Autore Mahnke Reinhard
Edizione [1st ed.]
Pubbl/distr/stampa Weinheim : , : John Wiley & Sons, Incorporated, , 2009
Descrizione fisica 1 online resource (450 pages)
Disciplina 519.23
Altri autori (Persone) KaupuzsJevgenijs
LubashevskyIhor
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.
ISBN 9783527626106
9783527408405
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
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.
Record Nr. UNINA-9910809145603321
Mahnke Reinhard  
Weinheim : , : John Wiley & Sons, Incorporated, , 2009
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui