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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Stochastic Numerical Methods introduces at Master level the numerical methods that use probability or stochastic concepts to analyze random processes. The book aims at being rather general and is addressed at students of natural sciences (Physics, Chemistry, Mathematics, Biology, etc.) and Engineering, but also social sciences (Economy, Sociology, etc.) where some of the techniques have been used recently to numerically simulate different agent-based models.
Examples included in the book range from phase-transitions and critical phenomena, including details of data analysis (extraction of critical exponents, finite-size effects, etc.), to population dynamics, interfacial growth, chemical reactions, etc. Program listings are integrated in the discussion of numerical algorithms to facilitate their understanding.
From the contents:
- Review of Probability Concepts
- Monte Carlo Integration
- Generation of Uniform and Non-uniform
- Random Numbers: Non-correlated Values
- Dynamical Methods
- Applications to Statistical Mechanics
- Introduction to Stochastic Processes
- Numerical Simulation of Ordinary and
- Partial Stochastic Differential Equations
- Introduction to Master Equations
- Numerical Simulations of Master Equations
- Hybrid Monte Carlo
- Generation of n-Dimensional Correlated
- Gaussian Variables
- Collective Algorithms for Spin Systems
- Histogram Extrapolation
- Multicanonical Simulations
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Seitenzahl: 624
Veröffentlichungsjahr: 2014
Ähnliche
Table of Contents
Cover
Related Titles
Title Page
Copyright
Dedication
Preface
Chapter 1: Review of probability concepts
1.1 Random Variables
1.2 Average Values, Moments
1.3 Some Important Probability Distributions with a Given Name
1.4 Successions of Random Variables
1.5 Jointly Gaussian Random Variables
1.6 Interpretation of the Variance: Statistical Errors
1.7 Sums of Random Variables
1.8 Conditional Probabilities
1.9 Markov Chains
Further Reading and References
Exercises
Chapter 2: Monte Carlo Integration
2.1 Hit and Miss
2.2 Uniform Sampling
2.3 General Sampling Methods
2.4 Generation of Nonuniform Random Numbers: Basic Concepts
2.5 Importance Sampling
2.6 Advantages of Monte Carlo Integration
2.7 Monte Carlo Importance Sampling for Sums
2.8 Efficiency of an Integration Method
2.9 Final Remarks
Further Reading and References
Exercises
Chapter 3: Generation of Nonuniform Random Numbers: Noncorrelated Values
3.1 General Method
3.2 Change of Variables
3.3 Combination of Variables
3.4 Multidimensional Distributions
3.5 Gaussian Distribution
3.6 Rejection Methods
Further Reading and References
Exercises
Chapter 4: Dynamical Methods
4.1 Rejection with Repetition: a Simple Case
4.2 Statistical Errors
4.3 Dynamical Methods
4.4 Metropolis
et al.
Algorithm
4.5 Multidimensional Distributions
4.6 Heat-Bath Method
4.7 Tuning the Algorithms
Further Reading and References
Exercises
Chapter 5: Applications to Statistical Mechanics
5.1 Introduction
5.2 Average Acceptance Probability
5.3 Interacting Particles
5.4 Ising Model
5.5 Heisenberg Model
5.6 Lattice Model
5.7 Data Analysis: Problems around the Critical Region
Further Reading and References
Exercises
Chapter 6: Introduction to Stochastic Processes
6.1 Brownian Motion
6.2 Stochastic Processes
6.3 Stochastic Differential Equations
6.4 White Noise
6.5 Stochastic Integrals. Itô and Stratonovich Interpretations
6.6 The Ornstein–Uhlenbeck Process
6.7 The Fokker–Planck Equation
Further Reading and References
Exercises
Chapter 7: Numerical Simulation of Stochastic Differential Equations
7.1 Numerical Integration of Stochastic Differential Equations with Gaussian White Noise
7.2 The Ornstein–Uhlenbeck Process: Exact Generation of Trajectories
7.3 Numerical Integration of Stochastic Differential Equations with Ornstein–Uhlenbeck Noise
7.4 Runge–Kutta-Type Methods
7.5 Numerical Integration of Stochastic Differential Equations with Several Variables
7.6 Rare Events: The Linear Equation with Linear Multiplicative Noise
7.7 First Passage Time Problems
7.8 Higher Order (?) Methods
Further Reading and References
Exercises
Chapter 8: Introduction to Master Equations
8.1 A Two-State System with Constant Rates
8.2 The General Case
8.3 Examples
8.4 The Generating Function Method for Solving Master Equations
8.5 The Mean-Field Theory
8.6 The Fokker–Planck Equation
Further Reading and References
Exercises
Chapter 9: Numerical Simulations of Master Equations
9.1 The First Reaction Method
9.2 The Residence Time Algorithm
Further Reading and References
Exercises
Chapter 10: Hybrid Monte Carlo
10.1 Molecular Dynamics
10.2 Hybrid Steps
10.3 Tuning of Parameters
10.4 Relation to Langevin Dynamics
10.5 Generalized Hybrid Monte Carlo
Further Reading and References
Exercises
Chapter 11: Stochastic Partial Differential Equations
11.1 Stochastic Partial Differential Equations
11.2 Coarse Graining
11.3 Finite Difference Methods for Stochastic Differential Equations
11.4 Time Discretization: von Neumann Stability Analysis
11.5 Pseudospectral Algorithms for Deterministic Partial Differential Equations
11.6 Pseudospectral Algorithms for Stochastic Differential Equations
11.7 Errors in the Pseudospectral Methods
Further Reading and References
Exercises
Appendix A: Generation of Uniform Û(0,1) Random Numbers
A.1 Pseudorandom Numbers
A.2 Congruential Generators
A.3 A Theorem by Marsaglia
A.4 Feedback Shift Register Generators
A.5 RCARRY and Lagged Fibonacci Generators
A.6 Final Advice
Exercises
Appendix B: Generation of n-Dimensional Correlated Gaussian Variables
B.1 The Gaussian Free Model
B.2 Translational Invariance
Exercises
Appendix C: Calculation of the Correlation Function of a Series
Exercises
Appendix D: Collective Algorithms for Spin Systems
Appendix E: Histogram Extrapolation
Appendix F: Multicanonical Simulations
Appendix G: Discrete Fourier Transform
G.1 Relation Between the Fourier Series and the Discrete Fourier Transform
G.2 Evaluation of Spatial Derivatives
G.3 The Fast Fourier Transform
Further Reading
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Chapter 1: Review of probability concepts
List of Illustrations
Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Figure 1.5
Figure 1.6
Figure 1.7
Figure 1.8
Figure 1.9
Figure 1.10
Figure 2.1
Figure 2.2
Figure 2.3
Figure 3.1
Figure 3.2
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5
Figure 9.1
Figure 9.2
Figure 9.3
Figure 9.4
Figure 9.5
Figure 11.1
Figure 11.6
Figure 11.2
Figure 11.3
Figure 11.4
Figure 11.5
Figure A.1
Figure B.1
Figure D.1
Figure E.1
Figure F.1
Figure G.1
Figure G.2
Figure G.3
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Raúl Toral and Pere Colet
Stochastic Numerical Methods
An Introduction for Students and Scientists
Authors
Prof. Raúl Toral
IFISC (Institute for Cross-disciplinary Physics and Complex Systems)
CSIC-Universitat Illes Balears
Palma de Mallorca
Spain
Prof. Pere Colet
IFISC (Institute for Cross-disciplinary Physics and Complex Systems)
CSIC-Universitat Illes Balears
Palma de Mallorca
Spain
Cover
The cover figure aims at exemplifying the random movement of Brownian particles in a potential landscape.
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.: applied for
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A catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978-3-527-41149-8
ePDF ISBN: 978-3-527-68313-0
ePub ISBN: 978-3-527-68312-3
Mobi ISBN: 978-3-527-68311-6
oBook ISBN: 978-3-527-68314-7
Dedicated to the memory of my father, from whom I learnt many important things not covered in this book.
Raúl Toral
Dedicated to my parents and to my wife for their support in different stages of my life.
Pere Colet
Preface
This book deals with numerical methods that use, in one way or another, concepts and ideas from probability theory and stochastic processes. This does not mean that their range of validity is limited to these fields, as there are many problems of a purely deterministic nature that can be tackled with these methods, among them, most noticeable, being the calculation of high-dimensional sums and integrals. The material covered in this book has grown out of a series of master courses taught by the authors in several universities and summer schools including, in particular, the Master in Physics of Complex Systems organized by IFISC (Instituto de Física Interdisciplinar y Systemas Complejos). It is aimed, then, at postgraduate students of almost any scientific discipline and also at those senior scientists who, not being familiar with the specificities of the numerical methods explained here, would like to get acquainted with the basic stochastic algorithms because they need them for a particular application in their field of research. The methods split naturally in three big blocks: sampling by Monte Carlo (Chapters 2–5 and 10), generation of trajectories of stochastic differential equations (Chapters 6, 7, and 11), and numerical solutions to master equations (Chapters 8 and 9), although they are intertwined in many occasions and we have tried to highlight those connections whenever they appear.
It has been our intention to keep the contents of the book self-contained. Hence, no previous knowledge of the subject is assumed by the reader. This is strictly true insofar as the numerical algorithms are concerned, but we have also included some chapters of a more theoretical nature where we summarize what the reader should know before facing the numerical chapters. These are: Chapter 1, a summary of probability concepts including the theory of Markov chains; Chapter 6, with a brief introduction to stochastic processes and stochastic differential equations, the concepts of white and colored noise, etc.; and Chapter 8, where we present briefly the master equations and some analytical tools for their analysis. The reader who is not familiar with any of these topics will find here the necessary theoretical concepts to understand the numerical methods, but if the reader wants to get a deeper knowledge of the theory, he or she might find it necessary to delve into the more advanced books mentioned in the bibliography section. Nevertheless, we have tried to keep the bibliographic references to a minimum, rather than include here and there numerous references to the different authors who have contributed to some aspect of the methods explained in the book. We feel that a book with a clear pedagogical orientation, as this one, is different from a review, and the reader should not be distracted by an excess ofreferences and we apologize to those authors who, having contributed to the field, do not find their work cited here. The basic bibliographic sites as well as suggestions for further reading have been included at the end of each chapter.
The goal of the book is to teach the reader to program the numerical algorithms to do different tasks. To this end, we have included more or less complete pieces of computer code. They are written in Fortran but they are, in the vast majority of cases, simple enough that they can be understood by anyone with a basic knowledge of programming in any language. The book is not intended to teach programming or code optimization. Sometimes we provide a full program, sometimes just some relevant lines of code. In any event, we do consider the lines of code to be part of the text, an important part that has to be read and analyzed in detail. Our experience indicates that full understanding is only achieved when one does not only know what he or she wants to do but also how this is implemented practically, and we do recommend the reader to implement and execute the computer programs along with the reading of the text. In particular, the section of the code that we consider absolutely essential has been framed in a box. It is not possible to reach a good understanding of the numerical algorithms without the understanding of the lines of code in the boxes.
We have included some exercises to complement the theory of the algorithms explained at the end of each chapter. We have not used the exercises to introduce difficult or more advanced topics but the exercises are, in general, of the level of the course and are given here so that the reader can test his or her level of comprehension of the algorithms and theory of the main text.
Some material is left for the appendices. They cover either standard material (generation of uniform random numbers, calculation of the correlation time of a series and discrete Fourier transforms), some more specialized topics (generation of Gaussian fields with a given correlation function, extrapolation techniques), or an introduction to more advanced simulation methods (collective algorithms, multicanonical simulations).
As we have said, the book is addressed to scientists of many disciplines, and beyond the general framework we have included only a few specific applications. In particular, in Chapter 5, we explain how to use the Monte Carlo sampling to derive properties of physical systems near a phase transition. The reason for the inclusion of this chapter is twofold. Historically, the field of phase transitions has made extensive use of the Monte Carlo methods (to the extent that some people might wrongly think that this is the only field of application). Moreover, it is the field of expertise of the authors, and we felt more confident explaining in detail how one can use the Monte Carlo sampling to derive precisely the equation of state and the critical exponents of some model systems of interest in Statistical Mechanics, including the Ising and scalar models, than other examples. Nevertheless, extensions of, for instance, the Ising model are now being used in fields as distant as sociology or economics, and we hope that the reader not particularly interested in the physical applications will still find some useful aspects of this chapter.
Finally, we would like to thank all the colleagues and students who have helped us to improve this book. In particular, Dr. Emilio Hernández–Garcia read and gave us valuable suggestions for Chapters 8 and 9.
1Review of probability concepts
In this chapter, we give a brief summary of the main concepts and results on probability and statistics which will be needed in the rest of the book. Readers who are familiar with the theory of probability might not need to read this chapter in detail, but we urge them to check that effectively this is the case.
1.1 Random Variables
In most occasions, we cannot predict with absolute certainty the outcome of an experiment (otherwise it might not be necessary to perform the experiment). We understand here the word “experiment” in a broad sense. We can count the number of electrons emitted by a β-radioactive substance in a given time interval, determine the time at which a projectile hits its target or a bus reaches the station, measure an electron's spin, toss a coin and look at the appearing side, or have a look through the window to observe whether it rains or not. We will denote by E the set of possible results of the experiment. For the β-radioactive substance, is the set of natural numbers ; the hitting times of the projectile or the arrival times of the bus (in some units) both belong to the set of real numbers ; the possible outcomes of a measure of an electron's spin are ; when tossing a dice, the possible results are ; and, finally, for the rain observation the set of results is . In all these cases, we have no way (or no effective way) of knowing a priori which one of the possible outcomes will be observed. Hence, we abandon the deterministic point of view and adopt a “probabilistic” description in which subsets of results (which are called “events”) are assigned a number measuring their likelihood of appearance. The “theory of probability” is the branch of mathematics that allows us to perform such an assignation in a logically consistent way and compatible with our intuition of how this likelihood of events should behave.
It is useful for the theory to consider that the set of results contains only numbers. In this way, we can use the rules of calculus (add, multiply, differentiate, integrate, etc.). If the results themselves are numbers (case of counting the number of electrons, determine the time of impact of the projectile, etc.), this requires no special consideration. In other cases (to observe whether it rains or not), we need to label each result with a number. This assignation is arbitrary, but usually it responds to some logics of the problem under consideration. For instance, when tossing a coin, it might be that we win every time heads show up and we lose when tails appear. The “natural” identification is for heads and for tails. This assignation of a number to the result of an experiment is called a “random variable.” Random variables are, hence, an application of the set of results to the set of real numbers. This application maps each result of the experiment into one, and only one, number. The application need not be one-to-one, but it can be many-to-one. For instance, if the experiment is to extract cards from a shuffled deck, we could assign to all hearts cards, to spades, and to diamonds and clubs. It is customary to denote random variables by using a “hat” on top of its name, say , or , or whatever name we choose for it. If we choose the name , the number associated with the result ξ is
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