| Description |
1 online resource (450 p.) |
| Series |
New York Academy of Sciences Ser |
|
New York Academy of Sciences Ser
|
| Contents |
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 |
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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 |
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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 |
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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 |
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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 |
| Notes |
Description based upon print version of record |
|
6.7 Summary of Results for Diffusion in a Finite Interval |
| Genre/Form |
Electronic books
|
| Form |
Electronic book
|
| ISBN |
9783527626106 |
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3527626107 |
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