Handbook of stochastic methods for physics, chemistry and the natural sciences:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1990
|
| Ausgabe: | 2. ed., corr. 2. print. |
| Schriftenreihe: | Springer series in synergetics
13 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XIX, 442 S. graph. Darst. |
| ISBN: | 3540156070 0387156070 |
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| 100 | 1 | |a Gardiner, Crispin W. |d 1942- |e Verfasser |0 (DE-588)108947959X |4 aut | |
| 245 | 1 | 0 | |a Handbook of stochastic methods for physics, chemistry and the natural sciences |c C. W. Gardiner |
| 250 | |a 2. ed., corr. 2. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1990 | |
| 300 | |a XIX, 442 S. |b graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Springer series in synergetics |v 13 | |
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Datensatz im Suchindex
| _version_ | 1804122756124508160 |
|---|---|
| adam_text | Contents
1. A Historical Introduction
1.1 Motivation 1
1.2 Some Historical Examples 2
1.2.1 Brownian Motion 2
1.2.2 Langevin s Equation 6
1.3 Birth-Death Processes 8
1.4 Noise in Electronic Systems 11
1.4.1 ShotNoise 11
1.4.2 Autocorrelation Functions and Spectra 15
1.4.3 Fourier Analysis of Fluctuating Functions:
Stationary Systems 17
1.4.4 Johnson Noise and Nyquist s Theorem 18
2. Probability Concepts
2.1 Events, and Sets of Events 21
2.2 Probabilities 22
2.2.1 Probability Axioms 22
2.2.2 The Meaning of P(A) 23
2.2.3 The Meaning of the Axioms 23
2.2.4 Random Variables 24
2.3 Joint and Conditional Probabilities: Independence 25
2.3.1 Joint Probabilities 25
2.3.2 Conditional Probabilities 25
2.3.3 Relationship Between Joint Probabilities of Different Orders 26
2.3.4 Independence 27
2.4 Mean Values and Probability Density 28
2.4.1 Determination of Probability Density by
Means of Arbitrary Functions 28
2.4.2 Sets of Probability Zero 29
2.5 Mean Values 29
2.5.1 Moments, Correlations, and Covariances 30
2.5.2 The Law of Large Numbers 30
2.6 Characteristic Function 32
2.7 Cumulant Generating Function:
Correlation Functions and Cumulants 33
2.7.1.Example:CumulantofOrder4:«^T1X2^ 3A 4» 35
2.7.2 Significance of Cumulants 35
XIV Contents
2.8 Gaussian and Poissonian Probability Distributions 36
2.8.1 The Gaussian Distribution 36
2.8.2 Central Limit Theorem 37
2.8.3 The Poisson Distribution 38
2.9 Limits of Sequences of Random Variables 39
2.9.1 Almost Certain Limit 40
2.9.2 Mean Square Limit (Limit in the Mean) 40
2.9.3 Stochastic Limit, or Limit in Probability 40
2.9.4 Limit in Distribution 41
2.9.5 Relationship Between Limits 41
3. Markov Processes
3.1 Stochastic Processes 42
3.2 Markov Process 43
3.2.1 Consistency — the Chapman-Kolmogorov Equation 43
3.2.2 Discrete State Spaces 44
3.2.3 More General Measures 44
3.3 Continuity in Stochastic Processes 45
3.3.1 Mathematical Definition of a Continuous Markov Process .. 46
3.4 Differential Chapman-Kolmogorov Equation 47
3.4.1 Derivation of the Differential
Chapman-Kolmogorov Equation 48
3.4.2 Status of the Differential Chapman-Kolmogorov Equation . 51
3.5 Interpretation of Conditions and Results 51
3.5.1 Jump Processes: The Master Equation 52
3.5.2 Diffusion Processes — the Fokker-Planck Equation 52
3.5.3 Deterministic Processes - Liouville s Equation 53
3.5.4 General Processes 54
3.6 Equations for Time Development in Initial Time -
Backward Equations 55
3.7 Stationary and Homogeneous Markov Processes 56
3.7.1 Ergodic Properties 57
3.7.2 Homogeneous Processes 60
3.7.3 Approach to a Stationary Process 61
3.7.4 Autocorrelation Function for Markov Processes 64
3.8 Examples of Markov Processes 66
3.8.1 The Wiener Process 66
3.8.2 The Random Walk in One Dimension 70
3.8.3 Poisson Process 73
3.8.4 The Ornstein-Uhlenbeck Process 74
3.8.5 Random Telegraph Process 77
4. The Ito Calculus and Stochastic Differentia] Equations
4.1 Motivation 80
4.2 Stochastic Integration 83
Contents XV
4.2.1 Definition of the Stochastic Integral 83
4.2.2 Example j W(t )dW(t ) 84
4.2.3 The Stratonovich Integral 86
4.2.4 Nonanticipating Functions 86
4.2.5 Proof that dW(t)2 = dt and dW{t)1+N =0 87
4.2.6 Properties of the Ito Stochastic Integral 88
4.3 Stochastic Differential Equations (SDE) 92
4.3.1 Ito Stochastic Differential Equation: Definition 93
4.3.2 Markov Property of the Solution of an
Ito Stochastic Differential Equation 95
4.3.3 Change of Variables: Ito s Formula 95
4.3.4 Connection Between Fokker-Planck Equation and
Stochastic Differential Equation 96
4.3.5 Multivariable Systems 97
4.3.6 Stratonovich s Stochastic Differential Equation 98
4.3.7 Dependence on Initial Conditions and Parameters 101
4.4 Some Examples and Solutions 102
4.4.1 Coefficients Without x Dependence 102
4.4.2 Multiplicative Linear White Noise Process 103
4.4.3 Complex Oscillator with Noisy Frequency 104
4.4.4 Ornstein-Uhlenbeck Process 106
4.4.5 Conversion from Cartesian to Polar Coordinates 107
4.4.6 Multivariate Ornstein-Uhlenbeck Process 109
4.4.7 The General Single Variable Linear Equation 112
4.4.8 Multivariable Linear Equations 114
4.4.9 Time-Dependent Ornstein-Uhlenbeck Process 115
5. The Fokker-Planck Equation
5.1 Background 117
5.2 Fokker-Planck Equation in One Dimension 118
5.2.1 Boundary Conditions 118
5.2.2 Stationary Solutions for Homogeneous Fokker-Planck
Equations 124
5.2.3 Examples of Stationary Solutions 126
5.2.4 Boundary Conditions for the Backward Fokker-Planck
Equation 128
5.2.5 Eigenfunction Methods (Homogeneous Processes) 129
5.2.6 Examples 132
5.2.7 First Passage Times for Homogeneous Processes 136
5.2.8 Probability of Exit Through a Particular End of the
Interval 142
5.3 Fokker-Planck Equations in Several Dimensions 143
5.3.1 Change of Variables 144
5.3.2 Boundary Conditions 146
5.3.3 Stationary Solutions: Potential Conditions 146
5.3.4 Detailed Balance 148
XVI Contents
5.3.5 Consequences of Detailed Balance 150
5.3.6 Examples of Detailed Balance in Fokker-Planck Equations . 155
5.3.7 Eigenfunction Methods in Many Variables -
Homogeneous Processes 165
5.4 First Exit Time from a Region (Homogeneous Processes) 170
5.4.1 Solutions of Mean Exit Time Problems 171
5.4.2 Distribution of Exit Points 174
6. Approximation Methods for Diffusion Processes
6.1 Small Noise Perturbation Theories 177
6.2 Small Noise Expansions for Stochastic Differential Equations .... 180
6.2.1 Validity of the Expansion 182
6.2.2 Stationary Solutions (Homogeneous Processes) 183
6.2.3 Mean, Variance, and Time Correlation Function 184
6.2.4 Failure of Small Noise Perturbation Theories 185
6.3 Small Noise Expansion of the Fokker-Planck Equation 187
6.3.1 Equations for Moments and Autocorrelation Functions .... 189
6.3.2 Example 192
6.3.3 Asymptotic Method for Stationary Distributions 194
6.4 Adiabatic Elimination of Fast Variables 195
6.4.1 Abstract Formulation in Terms of Operators
and Projectors 198
6.4.2 Solution Using Laplace Transform 200
6.4.3 Short-Time Behaviour 203
6.4.4 Boundary Conditions 205
6.4.5 Systematic Perturbative Analysis 206
6.5 White Noise Process as a Limit of Nonwhite Process 210
6.5.1 Generality of the Result 215
6.5.2 More General Fluctuation Equations 215
6.5.3 Time Nonhomogeneous Systems 216
6.5.4 Effect of Time Dependence in L, 217
6.6 Adiabatic Elimination of Fast Variables: The General Case 218
6.6.1 Example: Elimination of Short-Lived
Chemical Intermediates 218
6.6.2 Adiabatic Elimination in Haken s Model 223
6.6.3 Adiabatic Elimination of Fast Variables:
A Nonlinear Case 227
6.6.4 An Example with Arbitrary Nonlinear Coupling 232
7. Master Equations and Jump Processes
7.1 Birth-Death Master Equations - One Variable 236
7.1.1 Stationary Solutions 236
7.1.2 Example: Chemical Reaction X^A 238
7.1.3 A Chemical Bistable System 241
7.2 Approximation of Master Equations by Fokker-Planck Equations 246
7.2.1 Jump Process Approximation of a Diffusion Process 246
Contents XVII
7.2.2 The Kramers-Moyal Expansion 249
7.2.3 Van Kampen s System Size Expansion 250
7.2.4 Kurtz s Theorem 254
7.2.5 Critical Fluctuations 255
7.3 Boundary Conditions for Birth-Death Processes 257
7.4 Mean First Passage Times 259
7.4.1 Probability of Absorption 261
7.4.2 Comparison with Fokker-Planck Equation 261
7.5 Birth-Death Systems with Many Variables 262
7.5.1 Stationary Solutions when Detailed Balance Holds 263
7.5.2 Stationary Solutions Without Detailed Balance
(Kirchoff s Solution) 266
7.5.3 System Size Expansion and Related Expansions 266
7.6 Some Examples 267
7.6.1 X + A ^ 2X 267
7.6.2 X £ Y A A 267
k y
7.6.3 Prey-Predator System 268
7.6.4 Generating Function Equations 273
7.7 The Poisson Representation 277
7.7.1 Kinds of Poisson Representations 282
7.7.2 Real Poisson Representations 282
7.7.3 Complex Poisson Representations 282
7.7.4 The Positive Poisson Representation 285
7.7.5 Time Correlation Functions 289
7.7.6 Trimolecular Reaction 294
7.7.7. Third-Order Noise 299
8. Spatially Distributed Systems
8.1 Background 303
8.1.1 Functional Fokker-Planck Equations 305
8.2 Multivariate Master Equation Description 307
8.2.1 Diffusion 307
8.2.2 Continuum Form of Diffusion Master Equation 308
8.2.3 Reactions and Diffusion Combined 313
8.2.4 Poisson Representation Methods 314
8.3 Spatial and Temporal Correlation Structures 315
8.3.1 Reaction^ ^ Y 315
k2 k,
8.3.2 Reactions B + X ^ C, A + X -r 2X 319
8.3.3 A Nonlinear Model with a Second-Order Phase Transition .. 324
8.4 Connection Between Local and Global Descriptions 328
8.4.1 Explicit Adiabatic Elimination of Inhomogeneous Modes ... 328
8.5 Phase-Space Master Equation 331
8.5.1 Treatment of Flow 331
8.5.2 Flow as a Birth-Death Process 332
8.5.3 Inclusion of Collisions - the Boltzmann Master Equation .. 336
8.5.4 Collisions and Flow Together 339
XVIII Contents
9. Bistability, Metastability, and Escape Problems
9.1 Diffusion in a Double-Well Potential (One Variable) 342
9.1.1 Behaviour for£ = 0 343
9.1.2 Behaviour if D is Very Small 343
9.1.3 Exit Time 345
9.1.4 Splitting Probability 345
9.1.5 Decay from an Unstable State 347
9.2 Equilibration of Populations in Each Well 348
9.2.1 Kramers Method 349
9.2.2 Example: Reversible Denaturation of Chymotrypsinogen .. 352
9.2.3 Bistability with Birth-Death Master Equations
(One Variable) 354
9.3 Bistability in Multivariable Systems 357
9.3.1 Distribution of Exit Points 357
9.3.2 Asymptotic Analysis of Mean Exit Time 362
9.3.3 Kramers Method in Several Dimensions 363
9.3.4 Example: Brownian Motion in a Double Potential 366
10. Quantum Mechanical Markov Processes
10.1 Quantum Mechanics of the Harmonic Oscillator 373
10.1.1 Interaction with an External Field 375
10.1.2 Properties of Coherent States 376
10.2 Density Matrix and Probabilities 380
10.2.1 Von Neumann s Equation 382
10.2.2 Glauber-SudarshanP-Representation 382
10.2.3 Operator Correspondences 383
10.2.4 Application to the Driven Harmonic Oscillator 384
10.2.5 Quantum Characteristic Function 386
10.3 Quantum Markov Processes 388
10.3.1 Heat Bath 388
10.3.2 Correlations of Smooth Functions of Bath Operators ... 389
10.3.3 Quantum Master Equation for a System Interacting
with a Heat Bath 390
10.4 Examples and Applications of Quantum Markov Processes .... 395
10.4.1 Harmonic Oscillator 395
10.4.2 The Driven Two-Level Atom 399
10.5 Time Correlation Functions in Quantum Markov Processes .... 402
10.5.1 Quantum Regression Theorem 404
10.5.2 Application to Harmonic Oscillator
in the P-Representation 405
10.5.3 Time Correlations for Two-Level Atom 408
10.6 Generalised P-Representations 408
10.6.1 Definition of Generalised P-Representation 409
10.6.2 Existence Theorems 411
10.6.3 Relation to Poisson Representation 413
10.6.4 Operator Identities 414
Contents XIX
10.7 Application of Generalised P-Representations
to Time-Development Equations 415
10.7.1 Complex P-Representation 416
10.7.2 Positive P-Representation 416
10.7.3 Example 418
References 421
Bibliography 427
Symbol Index 431
Author Index 435
Subject Index 437
|
| any_adam_object | 1 |
| author | Gardiner, Crispin W. 1942- |
| author_GND | (DE-588)108947959X |
| author_facet | Gardiner, Crispin W. 1942- |
| author_role | aut |
| author_sort | Gardiner, Crispin W. 1942- |
| author_variant | c w g cw cwg |
| building | Verbundindex |
| bvnumber | BV008377689 |
| callnumber-first | Q - Science |
| callnumber-label | QA274 |
| callnumber-raw | QA274 |
| callnumber-search | QA274 |
| callnumber-sort | QA 3274 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 440 SK 820 SK 950 UG 3900 |
| classification_tum | PHY 015f MAT 606f |
| ctrlnum | (OCoLC)21219172 (DE-599)BVBBV008377689 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik Wirtschaftswissenschaften |
| edition | 2. ed., corr. 2. print. |
| format | Book |
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| id | DE-604.BV008377689 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:18:56Z |
| institution | BVB |
| isbn | 3540156070 0387156070 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005524844 |
| oclc_num | 21219172 |
| open_access_boolean | |
| owner | DE-473 DE-BY-UBG DE-20 DE-384 DE-188 DE-11 DE-706 DE-91G DE-BY-TUM |
| owner_facet | DE-473 DE-BY-UBG DE-20 DE-384 DE-188 DE-11 DE-706 DE-91G DE-BY-TUM |
| physical | XIX, 442 S. graph. Darst. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in synergetics |
| series2 | Springer series in synergetics |
| spelling | Gardiner, Crispin W. 1942- Verfasser (DE-588)108947959X aut Handbook of stochastic methods for physics, chemistry and the natural sciences C. W. Gardiner 2. ed., corr. 2. print. Berlin [u.a.] Springer 1990 XIX, 442 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in synergetics 13 Hier auch später erschienene, unveränderte Nachdrucke Stochastic processes Naturwissenschaften (DE-588)4041421-8 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Chemie (DE-588)4009816-3 gnd rswk-swf Synergetik (DE-588)4058755-1 gnd rswk-swf Stochastik (DE-588)4121729-9 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 s Mathematische Physik (DE-588)4037952-8 s DE-604 Synergetik (DE-588)4058755-1 s Stochastischer Prozess (DE-588)4057630-9 s 1\p DE-604 Naturwissenschaften (DE-588)4041421-8 s 2\p DE-604 Stochastik (DE-588)4121729-9 s 3\p DE-604 Chemie (DE-588)4009816-3 s 4\p DE-604 Physik (DE-588)4045956-1 s 5\p DE-604 Springer series in synergetics 13 (DE-604)BV000005271 13 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005524844&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 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Gardiner, Crispin W. 1942- Handbook of stochastic methods for physics, chemistry and the natural sciences Springer series in synergetics Stochastic processes Naturwissenschaften (DE-588)4041421-8 gnd Physik (DE-588)4045956-1 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Mathematische Physik (DE-588)4037952-8 gnd Chemie (DE-588)4009816-3 gnd Synergetik (DE-588)4058755-1 gnd Stochastik (DE-588)4121729-9 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| subject_GND | (DE-588)4041421-8 (DE-588)4045956-1 (DE-588)4057630-9 (DE-588)4037952-8 (DE-588)4009816-3 (DE-588)4058755-1 (DE-588)4121729-9 (DE-588)4132272-1 |
| title | Handbook of stochastic methods for physics, chemistry and the natural sciences |
| title_auth | Handbook of stochastic methods for physics, chemistry and the natural sciences |
| title_exact_search | Handbook of stochastic methods for physics, chemistry and the natural sciences |
| title_full | Handbook of stochastic methods for physics, chemistry and the natural sciences C. W. Gardiner |
| title_fullStr | Handbook of stochastic methods for physics, chemistry and the natural sciences C. W. Gardiner |
| title_full_unstemmed | Handbook of stochastic methods for physics, chemistry and the natural sciences C. W. Gardiner |
| title_short | Handbook of stochastic methods for physics, chemistry and the natural sciences |
| title_sort | handbook of stochastic methods for physics chemistry and the natural sciences |
| topic | Stochastic processes Naturwissenschaften (DE-588)4041421-8 gnd Physik (DE-588)4045956-1 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Mathematische Physik (DE-588)4037952-8 gnd Chemie (DE-588)4009816-3 gnd Synergetik (DE-588)4058755-1 gnd Stochastik (DE-588)4121729-9 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| topic_facet | Stochastic processes Naturwissenschaften Physik Stochastischer Prozess Mathematische Physik Chemie Synergetik Stochastik Stochastische Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005524844&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005271 |
| work_keys_str_mv | AT gardinercrispinw handbookofstochasticmethodsforphysicschemistryandthenaturalsciences |