The Fokker-Planck equation: methods of solution and applications
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1996
|
| Edition: | 2. ed., study ed. |
| Series: | Springer series in synergetics
18 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XIV, 472 S. graph. Darst. |
| ISBN: | 354061530X 9783540615309 3540504982 |
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MARC
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| 100 | 1 | |a Risken, Hannes |d 1934-1994 |e Verfasser |0 (DE-588)133955133 |4 aut | |
| 245 | 1 | 0 | |a The Fokker-Planck equation |b methods of solution and applications |c H. Risken |
| 250 | |a 2. ed., study ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1996 | |
| 300 | |a XIV, 472 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Springer series in synergetics |v 18 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
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| 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 |
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Record in the Search Index
| _version_ | 1804125395283345408 |
|---|---|
| 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
Itô 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 JV 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
Schrödinger
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), GOtO{t) and L0{t)
· · 259
10.4.2
Determination of Eigenvalues and Eigenvectors
........ 266
10.4.3
Expansion for the Green s Function Gn>m(/)
........... 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
χ
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
SolutionsWithNoise
.............................. 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
S.
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
|
| any_adam_object | 1 |
| author | Risken, Hannes 1934-1994 |
| author_GND | (DE-588)133955133 |
| author_facet | Risken, Hannes 1934-1994 |
| author_role | aut |
| author_sort | Risken, Hannes 1934-1994 |
| author_variant | h r hr |
| building | Verbundindex |
| bvnumber | BV010903747 |
| callnumber-first | Q - Science |
| callnumber-label | QC20 |
| callnumber-raw | QC20.7.D5 |
| callnumber-search | QC20.7.D5 |
| callnumber-sort | QC 220.7 D5 |
| callnumber-subject | QC - Physics |
| classification_rvk | QH 150 SK 950 UG 3900 SK 540 |
| classification_tum | PHY 013f MAT 356f PHY 064f |
| ctrlnum | (OCoLC)246734894 (DE-599)BVBBV010903747 |
| dewey-full | 530.1/3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.1/3 |
| dewey-search | 530.1/3 |
| dewey-sort | 3530.1 13 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik Wirtschaftswissenschaften |
| edition | 2. ed., study ed. |
| format | Book |
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| id | DE-604.BV010903747 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:00:53Z |
| institution | BVB |
| isbn | 354061530X 9783540615309 3540504982 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007293026 |
| oclc_num | 246734894 |
| open_access_boolean | |
| owner | DE-703 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-384 DE-355 DE-BY-UBR DE-83 DE-11 DE-188 DE-473 DE-BY-UBG DE-20 DE-824 DE-29T |
| owner_facet | DE-703 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-384 DE-355 DE-BY-UBR DE-83 DE-11 DE-188 DE-473 DE-BY-UBG DE-20 DE-824 DE-29T |
| physical | XIV, 472 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in synergetics |
| series2 | Springer series in synergetics |
| spelling | Risken, Hannes 1934-1994 Verfasser (DE-588)133955133 aut The Fokker-Planck equation methods of solution and applications H. Risken 2. ed., study ed. Berlin [u.a.] Springer 1996 XIV, 472 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in synergetics 18 Hier auch später erschienene, unveränderte Nachdrucke Statistische Thermodynamik (DE-588)4126251-7 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Fokker-Planck-Gleichung (DE-588)4126333-9 gnd rswk-swf Fokker-Planck-Gleichung (DE-588)4126333-9 s DE-604 Statistische Thermodynamik (DE-588)4126251-7 s 1\p DE-604 Stochastische Differentialgleichung (DE-588)4057621-8 s 2\p DE-604 Brownsche Bewegung (DE-588)4128328-4 s 3\p DE-604 Springer series in synergetics 18 (DE-604)BV000005271 18 Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007293026&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Risken, Hannes 1934-1994 The Fokker-Planck equation methods of solution and applications Springer series in synergetics Statistische Thermodynamik (DE-588)4126251-7 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Brownsche Bewegung (DE-588)4128328-4 gnd Fokker-Planck-Gleichung (DE-588)4126333-9 gnd |
| subject_GND | (DE-588)4126251-7 (DE-588)4057621-8 (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_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 | Statistische Thermodynamik (DE-588)4126251-7 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Brownsche Bewegung (DE-588)4128328-4 gnd Fokker-Planck-Gleichung (DE-588)4126333-9 gnd |
| topic_facet | Statistische Thermodynamik Stochastische Differentialgleichung Brownsche Bewegung Fokker-Planck-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007293026&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005271 |
| work_keys_str_mv | AT riskenhannes thefokkerplanckequationmethodsofsolutionandapplications |