The Fokker-Planck equation: methods of solution and applications
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1989
|
| Edition: | 2. ed. |
| Series: | Springer series in synergetics
18 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 472 S. |
| ISBN: | 3540504982 0387504982 |
Staff View
MARC
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| 100 | 1 | |a Risken, Hannes |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The Fokker-Planck equation |b methods of solution and applications |c H. Risken |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1989 | |
| 300 | |a XIV, 472 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Springer series in synergetics |v 18 | |
| 650 | 4 | |a Fokker-Planck, Équation de | |
| 650 | 7 | |a Fokker-Planck, équation de |2 ram | |
| 650 | 4 | |a Fokker-Planck equation | |
| 650 | 0 | 7 | |a Stochastische Differentialgleichung |0 (DE-588)4057621-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Statistische Thermodynamik |0 (DE-588)4126251-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Brownsche Bewegung |0 (DE-588)4128328-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Fokker-Planck-Gleichung |0 (DE-588)4126333-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Stochastische Differentialgleichung |0 (DE-588)4057621-8 |D s |
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Record in the Search Index
| _version_ | 1804136081440899072 |
|---|---|
| adam_text | Contents
1. Introduction 1
1.1 Brownian Motion 1
1.1.1 Deterministic Differential Equation 1
1.1.2 Stochastic Differential Equation 2
1.1.3 Equation of Motion for the Distribution Function 3
1.2 Fokker Planck Equation 4
1.2.1 Fokker Planck Equation for One Variable 4
1.2.2 Fokker Planck Equation for N Variables 5
1.2.3 How Does a Fokker Planck Equation Arise? 5
1.2.4 Purpose of the Fokker Planck Equation 6
1.2.5 Solutions of the Fokker Planck Equation 7
1.2.6 Kramers and Smoluchowski Equations 7
1.2.7 Generalizations of the Fokker Planck Equation 8
1.3 Boltzmann Equation 9
1.4 Master Equation 11
2. Probability Theory 13
2.1 Random Variable and Probability Density 13
2.2 Characteristic Function and Cumulants 16
2.3 Generalization to Several Random Variables 19
2.3.1 Conditional Probability Density 21
2.3.2 Cross Correlation 21
2.3.3 Gaussian Distribution 23
2.4 Time Dependent Random Variables 25
2.4.1 Classification of Stochastic Processes 26
2.4.2 Chapman Kolmogorov Equation 28
2.4.3 Wiener Khintchine Theorem 29
2.5 Several Time Dependent Random Variables 30
3. Langevin Equations 32
3.1 Langevin Equation for Brownian Motion 32
3.1.1 Mean Squared Displacement 34
3.1.2 Three Dimensional Case 36
3.1.3 Calculation of the Stationary Velocity Distribution Function 36
X Contents
3.2 Ornstein Uhlenbeck Process 38
3.2.1 Calculation of Moments 39
3.2.2 Correlation Function 41
3.2.3 Solution by Fourier Transformation 42
3.3 Nonlinear Langevin Equation, One Variable 44
3.3.1 Example 45
3.3.2 Kramers Moyal Expansion Coefficients 48
3.3.3 Ito s and Stratonovich s Definitions 50
3.4 Nonlinear Langevin Equations, Several Variables 54
3.4.1 Determination of the Langevin Equation from Drift and
Diffusion Coefficients 56
3.4.2 Transformation of Variables 57
3.4.3 How to Obtain Drift and Diffusion Coefficients for Systems 58
3.5 Markov Property 59
3.6 Solutions of the Langevin Equation by Computer Simulation .... 60
4. Fokker Planck Equation 63
4.1 Kramers Moyal Forward Expansion 63
4.1.1 Formal Solution 66
4.2 Kramers Moyal Backward Expansion 67
4.2.1 Formal Solution 69
4.2.2 Equivalence of the Solutions of the Forward and Backward
Equations 69
4.3 Pawula Theorem 70
4.4 Fokker Planck Equation for One Variable 72
4.4.1 Transition Probability Density for Small Times 73
4.4.2 Path Integral Solutions 74
4.5 Generation and Recombination Processes 76
4.6 Application of Truncated Kramers Moyal Expansions 77
4.7 Fokker Planck Equation for TV Variables 81
4.7.1 Probability Current 84
4.7.2 Joint Probability Distribution 85
4.7.3 Transition Probability Density for Small Times 85
4.8 Examples for Fokker Planck Equations with Several Variables ... 86
4.8.1 Three Dimensional Brownian Motion without Position
Variable 86
4.8.2 One Dimensional Brownian Motion in a Potential 87
4.8.3 Three Dimensional Brownian Motion in an External Force 87
4.8.4 Brownian Motion of Two Interacting Particles in an External
Potential 88
4.9 Transformation of Variables 88
4.10 Covariant Form of the Fokker Planck Equation 91
5. Fokker Planck Equation for One Variable; Methods of Solution 96
5.1 Normalization 96
5.2 Stationary Solution 98
Contents XI
5.3 Ornstein Uhlenbeck Process 99
5.4 Eigenfunction Expansion 101
5.5 Examples 108
5.5.1 Parabolic Potential 108
5.5.2 Inverted Parabolic Potential 109
5.5.3 Infinite Square Well for the SchrGdinger Potential 110
5.5.4 F Shaped Potential for the Fokker Planck Equation Ill
5.6 Jump Conditions 112
5.7 A Bistable Model Potential 114
5.8 Eigenfunctions and Eigenvalues of Inverted Potentials 117
5.9 Approximate and Numerical Methods for Determining
Eigenvalues and Eigenfunctions 119
5.9.1 Variational Method 120
5.9.2 Numerical Integration 120
5.9.3 Expansion into a Complete Set 121
5.10 Diffusion Over a Barrier 122
5.10.1 Kramers Escape Rate 123
5.10.2 Bistable and Metastable Potential 125
6. Fokker Planck Equation for Several Variables; Methods of Solution .. 133
6.1 Approach of the Solutions to a Limit Solution 134
6.2 Expansion into a Biorthogonal Set 137
6.3 Transformation of the Fokker Planck Operator, Eigenfunction
Expansions 139
6.4 Detailed Balance 145
6.5 Ornstein Uhlenbeck Process 153
6.6 Further Methods for Solving the Fokker Planck Equation 158
6.6.1 Transformation of Variables 158
6.6.2 Variational Method 158
6.6.3 Reduction to an Hermitian Problem 159
6.6.4 Numerical Integration 159
6.6.5 Expansion into Complete Sets 159
6.6.6 Matrix Continued Fraction Method 160
6.6.7 WKB Method 162
7. Linear Response and Correlation Functions 163
7.1 Linear Response Function 164
7.2 Correlation Functions 166
7.3 Susceptibility 172
8. Reduction of the Number of Variables 179
8.1 First Passage Time Problems 179
8.2 Drift and Diffusion Coefficients Independent of Some Variables 183
8.2.1 Time Integrals of Markovian Variables 184
XII Contents
8.3 Adiabatic Elimination of Fast Variables 188
8.3.1 Linear Process with Respect to the Fast Variable 192
8.3.2 Connection to the Nakajima Zwanzig Projector
Formalism 194
9. Solutions of Tridiagonal Recurrence Relations, Application to Ordinary
and Partial Differential Equations 196
9.1 Applications and Forms of Tridiagonal Recurrence Relations ... 197
9.1.1 Scalar Recurrence Relation 197
9.1.2 Vector Recurrence Relation 199
9.2 Solutions of Scalar Recurrence Relations 203
9.2.1 Stationary Solution 203
9.2.2 Initial Value Problem 209
9.2.3 Eigenvalue Problem 214
9.3 Solutions of Vector Recurrence Relations 216
9.3.1 Initial Value Problem 217
9.3.2 Eigenvalue Problem 220
9.4 Ordinary and Partial Differential Equations with Multiplicative
Harmonic Time Dependent Parameters 222
9.4.1 Ordinary Differential Equations 222
9.4.2 Partial Differential Equations 225
9.5 Methods for Calculating Continued Fractions 226
9.5.1 Ordinary Continued Fractions 226
9.5.2 Matrix Continued Fractions 227
10. Solutions of the Kramers Equation 229
10.1 Forms of the Kramers Equation 229
10.1.1 Normalization of Variables 230
10.1.2 Reversible and Irreversible Operators 231
10.1.3 Transformation of the Operators 233
10.1.4 Expansion into Hermite Functions 234
10.2 Solutions for a Linear Force 237
10.2.1 Transition Probability 238
10.2.2 Eigenvalues and Eigenfunctions 241
10.3 Matrix Continued Fraction Solutions of the Kramers Equation . 249
10.3.1 Initial Value Problem 251
10.3.2 Eigenvalue Problem 255
10.4 Inverse Friction Expansion 257
10.4.1 Inverse Friction Expansion for K0(t), G00(t) and L0(t) .. 259
10.4.2 Determination of Eigenvalues and Eigenvectors 266
10.4.3 Expansion for the Green s Function Gnm(t) 268
10.4.4 Position Dependent Friction 275
11. Brownian Motion in Periodic Potentials 276
11.1 Applications 280
11.1.1 Pendulum 280
Contents XIII
11.1.2 Superionic Conductor 280
11.1.3 Josephson Tunneling Junction 281
11.1.4 Rotation of Dipoles in a Constant Field 282
11.1.5 Phase Locked Loop 283
11.1.6 Connection to the Sine Gordon Equation 285
11.2 Normalization of the Langevin and Fokker Planck Equations .. 286
11.3 High Friction Limit 287
11.3.1 Stationary Solution 287
11.3.2 Time Dependent Solution 294
11.4 Low Friction Limit 300
11.4.1 Transformation to E and x Variables 301
11.4.2 Ansatz for the Stationary Distribution Functions 304
11.4.3 x Independent Functions 306
11.4.4 x Dependent Functions 307
11.4.5 Corrected x Independent Functions and Mobility 310
11.5 Stationary Solutions for Arbitrary Friction 314
11.5.1 Periodicity of the Stationary Distribution Function 315
11.5.2 Matrix Continued Fraction Method 317
11.5.3 Calculation of the Stationary Distribution Function .... 320
11.5.4 Alternative Matrix Continued Fraction for the Cosine
Potential 325
11.6 Bistability between Running and Locked Solution 328
11.6.1 Solutions Without Noise 329
11.6.2 Solutions With Noise 334
11.6.3 Low Friction Mobility With Noise 335
11.7 Instationary Solutions 337
11.7.1 Diffusion Constant 342
11.7.2 Transition Probability for Large Times 343
11.8 Susceptibilities 347
11.8.1 Zero Friction Limit 355
11.9 Eigenvalues and Eigenfunctions 359
11.9.1 Eigenvalues and Eigenfunctions in the Low Friction Limit 365
12. Statistical Properties of Laser Light 374
12.1 Semiclassical Laser Equations 377
12.1.1 Equations Without Noise 377
12.1.2 Langevin Equation 379
12.1.3 Laser Fokker Planck Equation 382
12.2 Stationary Solution and Its Expectation Values 384
12.3 Expansion in Eigenmodes 387
12.4 Expansion into a Complete Set; Solution by Matrix Continued
Fractions 394
12.4.1 Determination of Eigenvalues 396
12.5 Transient Solution 398
12.5.1 Eigenfunction Method 398
12.5.2 Expansion into a Complete Set 401
12.5.3 Solution for Large Pump Parameters 404
XIV Contents
12.6 Photoelectron Counting Distribution 408
12.6.1 Counting Distribution for Short Intervals 409
12.6.2 Expectation Values for Arbitrary Intervals 412
Appendices 414
Al Stochastic Differential Equations with Colored Gaussian Noise 414
A2 Boltzmann Equation with BGK and SW Collision Operators ... 420
A3 Evaluation of a Matrix Continued Fraction for the Harmonic
Oscillator 422
A4 Damped Quantum Mechanical Harmonic Oscillator 425
A5 Alternative Derivation of the Fokker Planck Equation 429
A6 Fluctuating Control Parameter 431
S. Supplement to the Second Edition 436
5.1 Solutions of the Fokker Planck Equation by Computer
Simulation (Sect. 3.6) 436
5.2 Kramers Moyal Expansion (Sect. 4.6) 436
5.3 Example for the Covariant Form of the Fokker Planck Equation
(Sect. 4.10) 437
5.4 Connection to Supersymmetry and Exact Solutions of the
One Variable Fokker Planck Equation (Chap. 5) 438
5.5 Nondifferentiability of the Potential for the Weak Noise
Expansion (Sects. 6.6 and 6.7) 438
5.6 Further Applications of Matrix Continued Fractions
(Chap. 9) 439
5.7 Brownian Motion in a Double Well Potential
(Chaps. 10 and 11) 439
5.8 Boundary Layer Theory (Sect. 11.4) 440
5.9 Calculation of Correlation Times (Sect. 7.12) 441
5.10 Colored Noise (Appendix Al) 443
5.11 Fokker Planck Equation with a Non Positive Definite Diffusion
Matrix and Fokker Planck Equation with Additional Third
Order Derivative Terms 445
References 448
Subject Index 463
|
| adam_txt |
Contents
1. Introduction 1
1.1 Brownian Motion 1
1.1.1 Deterministic Differential Equation 1
1.1.2 Stochastic Differential Equation 2
1.1.3 Equation of Motion for the Distribution Function 3
1.2 Fokker Planck Equation 4
1.2.1 Fokker Planck Equation for One Variable 4
1.2.2 Fokker Planck Equation for N Variables 5
1.2.3 How Does a Fokker Planck Equation Arise? 5
1.2.4 Purpose of the Fokker Planck Equation 6
1.2.5 Solutions of the Fokker Planck Equation 7
1.2.6 Kramers and Smoluchowski Equations 7
1.2.7 Generalizations of the Fokker Planck Equation 8
1.3 Boltzmann Equation 9
1.4 Master Equation 11
2. Probability Theory 13
2.1 Random Variable and Probability Density 13
2.2 Characteristic Function and Cumulants 16
2.3 Generalization to Several Random Variables 19
2.3.1 Conditional Probability Density 21
2.3.2 Cross Correlation 21
2.3.3 Gaussian Distribution 23
2.4 Time Dependent Random Variables 25
2.4.1 Classification of Stochastic Processes 26
2.4.2 Chapman Kolmogorov Equation 28
2.4.3 Wiener Khintchine Theorem 29
2.5 Several Time Dependent Random Variables 30
3. Langevin Equations 32
3.1 Langevin Equation for Brownian Motion 32
3.1.1 Mean Squared Displacement 34
3.1.2 Three Dimensional Case 36
3.1.3 Calculation of the Stationary Velocity Distribution Function 36
X Contents
3.2 Ornstein Uhlenbeck Process 38
3.2.1 Calculation of Moments 39
3.2.2 Correlation Function 41
3.2.3 Solution by Fourier Transformation 42
3.3 Nonlinear Langevin Equation, One Variable 44
3.3.1 Example 45
3.3.2 Kramers Moyal Expansion Coefficients 48
3.3.3 Ito's and Stratonovich's Definitions 50
3.4 Nonlinear Langevin Equations, Several Variables 54
3.4.1 Determination of the Langevin Equation from Drift and
Diffusion Coefficients 56
3.4.2 Transformation of Variables 57
3.4.3 How to Obtain Drift and Diffusion Coefficients for Systems 58
3.5 Markov Property 59
3.6 Solutions of the Langevin Equation by Computer Simulation . 60
4. Fokker Planck Equation 63
4.1 Kramers Moyal Forward Expansion 63
4.1.1 Formal Solution 66
4.2 Kramers Moyal Backward Expansion 67
4.2.1 Formal Solution 69
4.2.2 Equivalence of the Solutions of the Forward and Backward
Equations 69
4.3 Pawula Theorem 70
4.4 Fokker Planck Equation for One Variable 72
4.4.1 Transition Probability Density for Small Times 73
4.4.2 Path Integral Solutions 74
4.5 Generation and Recombination Processes 76
4.6 Application of Truncated Kramers Moyal Expansions 77
4.7 Fokker Planck Equation for TV Variables 81
4.7.1 Probability Current 84
4.7.2 Joint Probability Distribution 85
4.7.3 Transition Probability Density for Small Times 85
4.8 Examples for Fokker Planck Equations with Several Variables . 86
4.8.1 Three Dimensional Brownian Motion without Position
Variable 86
4.8.2 One Dimensional Brownian Motion in a Potential 87
4.8.3 Three Dimensional Brownian Motion in an External Force 87
4.8.4 Brownian Motion of Two Interacting Particles in an External
Potential 88
4.9 Transformation of Variables 88
4.10 Covariant Form of the Fokker Planck Equation 91
5. Fokker Planck Equation for One Variable; Methods of Solution 96
5.1 Normalization 96
5.2 Stationary Solution 98
Contents XI
5.3 Ornstein Uhlenbeck Process 99
5.4 Eigenfunction Expansion 101
5.5 Examples 108
5.5.1 Parabolic Potential 108
5.5.2 Inverted Parabolic Potential 109
5.5.3 Infinite Square Well for the SchrGdinger Potential 110
5.5.4 F Shaped Potential for the Fokker Planck Equation Ill
5.6 Jump Conditions 112
5.7 A Bistable Model Potential 114
5.8 Eigenfunctions and Eigenvalues of Inverted Potentials 117
5.9 Approximate and Numerical Methods for Determining
Eigenvalues and Eigenfunctions 119
5.9.1 Variational Method 120
5.9.2 Numerical Integration 120
5.9.3 Expansion into a Complete Set 121
5.10 Diffusion Over a Barrier 122
5.10.1 Kramers' Escape Rate 123
5.10.2 Bistable and Metastable Potential 125
6. Fokker Planck Equation for Several Variables; Methods of Solution . 133
6.1 Approach of the Solutions to a Limit Solution 134
6.2 Expansion into a Biorthogonal Set 137
6.3 Transformation of the Fokker Planck Operator, Eigenfunction
Expansions 139
6.4 Detailed Balance 145
6.5 Ornstein Uhlenbeck Process 153
6.6 Further Methods for Solving the Fokker Planck Equation 158
6.6.1 Transformation of Variables 158
6.6.2 Variational Method 158
6.6.3 Reduction to an Hermitian Problem 159
6.6.4 Numerical Integration 159
6.6.5 Expansion into Complete Sets 159
6.6.6 Matrix Continued Fraction Method 160
6.6.7 WKB Method 162
7. Linear Response and Correlation Functions 163
7.1 Linear Response Function 164
7.2 Correlation Functions 166
7.3 Susceptibility 172
8. Reduction of the Number of Variables 179
8.1 First Passage Time Problems 179
8.2 Drift and Diffusion Coefficients Independent of Some Variables 183
8.2.1 Time Integrals of Markovian Variables 184
XII Contents
8.3 Adiabatic Elimination of Fast Variables 188
8.3.1 Linear Process with Respect to the Fast Variable 192
8.3.2 Connection to the Nakajima Zwanzig Projector
Formalism 194
9. Solutions of Tridiagonal Recurrence Relations, Application to Ordinary
and Partial Differential Equations 196
9.1 Applications and Forms of Tridiagonal Recurrence Relations . 197
9.1.1 Scalar Recurrence Relation 197
9.1.2 Vector Recurrence Relation 199
9.2 Solutions of Scalar Recurrence Relations 203
9.2.1 Stationary Solution 203
9.2.2 Initial Value Problem 209
9.2.3 Eigenvalue Problem 214
9.3 Solutions of Vector Recurrence Relations 216
9.3.1 Initial Value Problem 217
9.3.2 Eigenvalue Problem 220
9.4 Ordinary and Partial Differential Equations with Multiplicative
Harmonic Time Dependent Parameters 222
9.4.1 Ordinary Differential Equations 222
9.4.2 Partial Differential Equations 225
9.5 Methods for Calculating Continued Fractions 226
9.5.1 Ordinary Continued Fractions 226
9.5.2 Matrix Continued Fractions 227
10. Solutions of the Kramers Equation 229
10.1 Forms of the Kramers Equation 229
10.1.1 Normalization of Variables 230
10.1.2 Reversible and Irreversible Operators 231
10.1.3 Transformation of the Operators 233
10.1.4 Expansion into Hermite Functions 234
10.2 Solutions for a Linear Force 237
10.2.1 Transition Probability 238
10.2.2 Eigenvalues and Eigenfunctions 241
10.3 Matrix Continued Fraction Solutions of the Kramers Equation . 249
10.3.1 Initial Value Problem 251
10.3.2 Eigenvalue Problem 255
10.4 Inverse Friction Expansion 257
10.4.1 Inverse Friction Expansion for K0(t), G00(t) and L0(t) . 259
10.4.2 Determination of Eigenvalues and Eigenvectors 266
10.4.3 Expansion for the Green's Function Gnm(t) 268
10.4.4 Position Dependent Friction 275
11. Brownian Motion in Periodic Potentials 276
11.1 Applications 280
11.1.1 Pendulum 280
Contents XIII
11.1.2 Superionic Conductor 280
11.1.3 Josephson Tunneling Junction 281
11.1.4 Rotation of Dipoles in a Constant Field 282
11.1.5 Phase Locked Loop 283
11.1.6 Connection to the Sine Gordon Equation 285
11.2 Normalization of the Langevin and Fokker Planck Equations . 286
11.3 High Friction Limit 287
11.3.1 Stationary Solution 287
11.3.2 Time Dependent Solution 294
11.4 Low Friction Limit 300
11.4.1 Transformation to E and x Variables 301
11.4.2 'Ansatz' for the Stationary Distribution Functions 304
11.4.3 x Independent Functions 306
11.4.4 x Dependent Functions 307
11.4.5 Corrected x Independent Functions and Mobility 310
11.5 Stationary Solutions for Arbitrary Friction 314
11.5.1 Periodicity of the Stationary Distribution Function 315
11.5.2 Matrix Continued Fraction Method 317
11.5.3 Calculation of the Stationary Distribution Function . 320
11.5.4 Alternative Matrix Continued Fraction for the Cosine
Potential 325
11.6 Bistability between Running and Locked Solution 328
11.6.1 Solutions Without Noise 329
11.6.2 Solutions With Noise 334
11.6.3 Low Friction Mobility With Noise 335
11.7 Instationary Solutions 337
11.7.1 Diffusion Constant 342
11.7.2 Transition Probability for Large Times 343
11.8 Susceptibilities 347
11.8.1 Zero Friction Limit 355
11.9 Eigenvalues and Eigenfunctions 359
11.9.1 Eigenvalues and Eigenfunctions in the Low Friction Limit 365
12. Statistical Properties of Laser Light 374
12.1 Semiclassical Laser Equations 377
12.1.1 Equations Without Noise 377
12.1.2 Langevin Equation 379
12.1.3 Laser Fokker Planck Equation 382
12.2 Stationary Solution and Its Expectation Values 384
12.3 Expansion in Eigenmodes 387
12.4 Expansion into a Complete Set; Solution by Matrix Continued
Fractions 394
12.4.1 Determination of Eigenvalues 396
12.5 Transient Solution 398
12.5.1 Eigenfunction Method 398
12.5.2 Expansion into a Complete Set 401
12.5.3 Solution for Large Pump Parameters 404
XIV Contents
12.6 Photoelectron Counting Distribution 408
12.6.1 Counting Distribution for Short Intervals 409
12.6.2 Expectation Values for Arbitrary Intervals 412
Appendices 414
Al Stochastic Differential Equations with Colored Gaussian Noise 414
A2 Boltzmann Equation with BGK and SW Collision Operators . 420
A3 Evaluation of a Matrix Continued Fraction for the Harmonic
Oscillator 422
A4 Damped Quantum Mechanical Harmonic Oscillator 425
A5 Alternative Derivation of the Fokker Planck Equation 429
A6 Fluctuating Control Parameter 431
S. Supplement to the Second Edition 436
5.1 Solutions of the Fokker Planck Equation by Computer
Simulation (Sect. 3.6) 436
5.2 Kramers Moyal Expansion (Sect. 4.6) 436
5.3 Example for the Covariant Form of the Fokker Planck Equation
(Sect. 4.10) 437
5.4 Connection to Supersymmetry and Exact Solutions of the
One Variable Fokker Planck Equation (Chap. 5) 438
5.5 Nondifferentiability of the Potential for the Weak Noise
Expansion (Sects. 6.6 and 6.7) 438
5.6 Further Applications of Matrix Continued Fractions
(Chap. 9) 439
5.7 Brownian Motion in a Double Well Potential
(Chaps. 10 and 11) 439
5.8 Boundary Layer Theory (Sect. 11.4) ' 440
5.9 Calculation of Correlation Times (Sect. 7.12) 441
5.10 Colored Noise (Appendix Al) 443
5.11 Fokker Planck Equation with a Non Positive Definite Diffusion
Matrix and Fokker Planck Equation with Additional Third
Order Derivative Terms 445
References 448
Subject Index 463 |
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| author | Risken, Hannes |
| author_facet | Risken, Hannes |
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| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV022106972 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:15:34Z |
| indexdate | 2024-07-09T20:50:44Z |
| institution | BVB |
| isbn | 3540504982 0387504982 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015321849 |
| oclc_num | 19127283 |
| open_access_boolean | |
| owner | DE-706 DE-83 DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-703 DE-739 DE-355 DE-BY-UBR DE-384 DE-29T DE-634 DE-188 DE-473 DE-BY-UBG DE-11 DE-20 |
| owner_facet | DE-706 DE-83 DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-703 DE-739 DE-355 DE-BY-UBR DE-384 DE-29T DE-634 DE-188 DE-473 DE-BY-UBG DE-11 DE-20 |
| physical | XIV, 472 S. |
| publishDate | 1989 |
| publishDateSearch | 1989 |
| publishDateSort | 1989 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in synergetics |
| series2 | Springer series in synergetics |
| spelling | Risken, Hannes Verfasser aut The Fokker-Planck equation methods of solution and applications H. Risken 2. ed. Berlin [u.a.] Springer 1989 XIV, 472 S. txt rdacontent n rdamedia nc rdacarrier Springer series in synergetics 18 Fokker-Planck, Équation de Fokker-Planck, équation de ram Fokker-Planck equation Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Statistische Thermodynamik (DE-588)4126251-7 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Fokker-Planck-Gleichung (DE-588)4126333-9 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s DE-604 Statistische Thermodynamik (DE-588)4126251-7 s Brownsche Bewegung (DE-588)4128328-4 s Fokker-Planck-Gleichung (DE-588)4126333-9 s 1\p DE-604 Springer series in synergetics 18 (DE-604)BV000005271 18 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015321849&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Risken, Hannes The Fokker-Planck equation methods of solution and applications Springer series in synergetics Fokker-Planck, Équation de Fokker-Planck, équation de ram Fokker-Planck equation Stochastische Differentialgleichung (DE-588)4057621-8 gnd Statistische Thermodynamik (DE-588)4126251-7 gnd Brownsche Bewegung (DE-588)4128328-4 gnd Fokker-Planck-Gleichung (DE-588)4126333-9 gnd |
| subject_GND | (DE-588)4057621-8 (DE-588)4126251-7 (DE-588)4128328-4 (DE-588)4126333-9 |
| title | The Fokker-Planck equation methods of solution and applications |
| title_auth | The Fokker-Planck equation methods of solution and applications |
| title_exact_search | The Fokker-Planck equation methods of solution and applications |
| title_exact_search_txtP | The Fokker-Planck equation methods of solution and applications |
| title_full | The Fokker-Planck equation methods of solution and applications H. Risken |
| title_fullStr | The Fokker-Planck equation methods of solution and applications H. Risken |
| title_full_unstemmed | The Fokker-Planck equation methods of solution and applications H. Risken |
| title_short | The Fokker-Planck equation |
| title_sort | the fokker planck equation methods of solution and applications |
| title_sub | methods of solution and applications |
| topic | Fokker-Planck, Équation de Fokker-Planck, équation de ram Fokker-Planck equation Stochastische Differentialgleichung (DE-588)4057621-8 gnd Statistische Thermodynamik (DE-588)4126251-7 gnd Brownsche Bewegung (DE-588)4128328-4 gnd Fokker-Planck-Gleichung (DE-588)4126333-9 gnd |
| topic_facet | Fokker-Planck, Équation de Fokker-Planck, équation de Fokker-Planck equation Stochastische Differentialgleichung Statistische Thermodynamik Brownsche Bewegung Fokker-Planck-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015321849&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005271 |
| work_keys_str_mv | AT riskenhannes thefokkerplanckequationmethodsofsolutionandapplications |