Stochastic tools in mathematics and science:
This introductory book on probability-based modeling covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. The topics include conditional expectations, stochastic processes, Brownian motion and its relation to partial differential equations, Langevin equations,...
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2006
|
| Series: | Surveys and tutorials in the applied mathematical sciences
1 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Summary: | This introductory book on probability-based modeling covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. The topics include conditional expectations, stochastic processes, Brownian motion and its relation to partial differential equations, Langevin equations, the Liouville and Fokker-Planck equations, as well as Markov chain Monte Carlo algorithms, renormalization and dimensional reduction, and basic equilibrium and non-equilibrium statistical mechanics. The applications include data assimilation, prediction from partial data, spectral analysis, and turbulence. A noteworthy feature of the book is the systematic analysis of memory effects. |
| Physical Description: | VIII, 147 S. graph. Darst. |
| ISBN: | 0387280804 9780387280806 0387280812 |
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Record in the Search Index
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| adam_text | Contents
Preface v
Chapter 1. Preliminaries 1
1.1. Least Squares Approximation 1
1.2. Orthonormal Bases 7
1.3. Fourier Series 9
1.4. Fourier Transform 12
1.5. Exercises Ki
1.6. Bibliography 1(5
Chapter 2. Probability 17
2.1. Definitions 17
2.2. Expected Values and Moments 20
2.3. Monte Carlo Methods 26
2.4. Parametric Estimation 29
2.5. The Central Limit Theorem 31
2.6. Conditional Probability and Conditional Expectation 34
2.7. Bayes Theorem 38
2.8. Exercises 40
2.9. Bibliography 42
Chapter 3. Brownian Motion 43
3.1. Definition of Brownian Motion 43
3.2. Brownian Motion and the Heat Equation 45
3.3. Solution of the Heat Equation by Random Walks 47
3.4. The Wiener Measure 50
3.5. Heat Equation with Potential 52
3.6. Physicists Notation for Wiener Measure 55
3.7. Another Connection Between Brownian Motion and the
Heat Equation 57
3.8. First Discussion of the Langevin Equation 59
3.9. Solution of a Nonlinear Differential Equation by Branching
Brownian Motion (i4
3.10. A Brief Introduction to Stochastic ODEs 05
3.11. Exercises 67
vii
viii CONTENTS
3.12. Bibliography 69
Chapter 4. Stationary Stochastic Processes 71
4.1. Weak Definition of a Stochastic Process 71
4.2. Covariance and Spectrum 74
4.3. The Inertial Spectrum of Turbulence 76
4.4. Random Measures and Random Fourier Transforms 78
4.5. Prediction for Stationary Stochastic Processes 84
4.6. Data Assimilation 89
4.7. Exercises 92
4.8. Bibliography 95
Chapter 5. Statistical Mechanics 97
5.1. Mechanics 97
5.2. Statistical Mechanics 99
5.3. Entropy and Equilibrium 102
5.4. The Ising Model 105
5.5. Markov Chain Monte Carlo 107
5.6. Renormalization 111
5.7. Exercises 116
5.8. Bibliography 117
Chapter 6. Time Dependent Statistical Mechanics 119
6.1. More on the Langevin Equation 119
6.2. A Coupled System of Harmonic Oscillators 122
6.3. Mathematical Addenda 124
6.4. The Mori Zwanzig Formalism 129
6.5. More on Fluctuation Dissipation Theorems 134
6.6. Scale Separation and Weak Coupling 135
6.7. Noninstantaneous Memory 137
6.8. Exercises 141
6.9. Bibliography 142
Index 145
|
| adam_txt |
Contents
Preface v
Chapter 1. Preliminaries 1
1.1. Least Squares Approximation 1
1.2. Orthonormal Bases 7
1.3. Fourier Series 9
1.4. Fourier Transform 12
1.5. Exercises Ki
1.6. Bibliography 1(5
Chapter 2. Probability 17
2.1. Definitions 17
2.2. Expected Values and Moments 20
2.3. Monte Carlo Methods 26
2.4. Parametric Estimation 29
2.5. The Central Limit Theorem 31
2.6. Conditional Probability and Conditional Expectation 34
2.7. Bayes' Theorem 38
2.8. Exercises 40
2.9. Bibliography 42
Chapter 3. Brownian Motion 43
3.1. Definition of Brownian Motion 43
3.2. Brownian Motion and the Heat Equation 45
3.3. Solution of the Heat Equation by Random Walks 47
3.4. The Wiener Measure 50
3.5. Heat Equation with Potential 52
3.6. Physicists' Notation for Wiener Measure 55
3.7. Another Connection Between Brownian Motion and the
Heat Equation 57
3.8. First Discussion of the Langevin Equation 59
3.9. Solution of a Nonlinear Differential Equation by Branching
Brownian Motion (i4
3.10. A Brief Introduction to Stochastic ODEs 05
3.11. Exercises 67
vii
viii CONTENTS
3.12. Bibliography 69
Chapter 4. Stationary Stochastic Processes 71
4.1. Weak Definition of a Stochastic Process 71
4.2. Covariance and Spectrum 74
4.3. The Inertial Spectrum of Turbulence 76
4.4. Random Measures and Random Fourier Transforms 78
4.5. Prediction for Stationary Stochastic Processes 84
4.6. Data Assimilation 89
4.7. Exercises 92
4.8. Bibliography 95
Chapter 5. Statistical Mechanics 97
5.1. Mechanics 97
5.2. Statistical Mechanics 99
5.3. Entropy and Equilibrium 102
5.4. The Ising Model 105
5.5. Markov Chain Monte Carlo 107
5.6. Renormalization 111
5.7. Exercises 116
5.8. Bibliography 117
Chapter 6. Time Dependent Statistical Mechanics 119
6.1. More on the Langevin Equation 119
6.2. A Coupled System of Harmonic Oscillators 122
6.3. Mathematical Addenda 124
6.4. The Mori Zwanzig Formalism 129
6.5. More on Fluctuation Dissipation Theorems 134
6.6. Scale Separation and Weak Coupling 135
6.7. Noninstantaneous Memory 137
6.8. Exercises 141
6.9. Bibliography 142
Index 145 |
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| index_date | 2024-07-02T14:12:05Z |
| indexdate | 2024-07-09T20:36:57Z |
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| spelling | Chorin, Alexandre Joel Verfasser (DE-588)109435303 aut Stochastic tools in mathematics and science Alexandre J. Chorin ; Ole H. Hald New York, NY Springer 2006 VIII, 147 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Surveys and tutorials in the applied mathematical sciences 1 This introductory book on probability-based modeling covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. The topics include conditional expectations, stochastic processes, Brownian motion and its relation to partial differential equations, Langevin equations, the Liouville and Fokker-Planck equations, as well as Markov chain Monte Carlo algorithms, renormalization and dimensional reduction, and basic equilibrium and non-equilibrium statistical mechanics. The applications include data assimilation, prediction from partial data, spectral analysis, and turbulence. A noteworthy feature of the book is the systematic analysis of memory effects. Mecânica dos fluídos computacional larpcal Natuurwetenschappen gtt Processos estocásticos larpcal Processus stochastiques Stochastische methoden gtt Toepassingen gtt Naturwissenschaft Stochastic processes Stochastik (DE-588)4121729-9 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Angewandte Mathematik (DE-588)4142443-8 gnd rswk-swf Stochastik (DE-588)4121729-9 s DE-604 Angewandte Mathematik (DE-588)4142443-8 s Stochastisches Modell (DE-588)4057633-4 s Hald, Ole H. Verfasser (DE-588)139218939 aut Surveys and tutorials in the applied mathematical sciences 1 (DE-604)BV021500680 1 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2667970&prov=M&dok_var=1&dok_ext=htm Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014705711&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chorin, Alexandre Joel Hald, Ole H. Stochastic tools in mathematics and science Surveys and tutorials in the applied mathematical sciences Mecânica dos fluídos computacional larpcal Natuurwetenschappen gtt Processos estocásticos larpcal Processus stochastiques Stochastische methoden gtt Toepassingen gtt Naturwissenschaft Stochastic processes Stochastik (DE-588)4121729-9 gnd Stochastisches Modell (DE-588)4057633-4 gnd Angewandte Mathematik (DE-588)4142443-8 gnd |
| subject_GND | (DE-588)4121729-9 (DE-588)4057633-4 (DE-588)4142443-8 |
| title | Stochastic tools in mathematics and science |
| title_auth | Stochastic tools in mathematics and science |
| title_exact_search | Stochastic tools in mathematics and science |
| title_exact_search_txtP | Stochastic tools in mathematics and science |
| title_full | Stochastic tools in mathematics and science Alexandre J. Chorin ; Ole H. Hald |
| title_fullStr | Stochastic tools in mathematics and science Alexandre J. Chorin ; Ole H. Hald |
| title_full_unstemmed | Stochastic tools in mathematics and science Alexandre J. Chorin ; Ole H. Hald |
| title_short | Stochastic tools in mathematics and science |
| title_sort | stochastic tools in mathematics and science |
| topic | Mecânica dos fluídos computacional larpcal Natuurwetenschappen gtt Processos estocásticos larpcal Processus stochastiques Stochastische methoden gtt Toepassingen gtt Naturwissenschaft Stochastic processes Stochastik (DE-588)4121729-9 gnd Stochastisches Modell (DE-588)4057633-4 gnd Angewandte Mathematik (DE-588)4142443-8 gnd |
| topic_facet | Mecânica dos fluídos computacional Natuurwetenschappen Processos estocásticos Processus stochastiques Stochastische methoden Toepassingen Naturwissenschaft Stochastic processes Stochastik Stochastisches Modell Angewandte Mathematik |
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