Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences PDF
Preview Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences
SPRINGER ass erleas IN SYNERGETICS COMPLE C. W. Gardiner Handbook e i l e LIKale m e c i y U m Methods for Physics, Chemistry and the Natural Sciences Cs Saba Springer Complexity Springer Complexity is a publication program,cutting acrossall traditional dis- ciplines ofsciences as well as engineering, economics, medicine, psychology and computersciences,whichisaimedatresearchers,studentsandpractitionerswork- ing in the field ofcomplex systems. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality ofmacroscopic collective behavior through self-organization,e.g., the spontaneous formation of temporal, spatial orfunctional structures. This recognition, thatthe collectivebe- havior ofthe whole system cannot be simply inferred from the understanding of the behavior ofthe individual components, hasled to various new concepts and sophisticated tools ofcomplexity. The main concepts andtools — with sometimes overlapping contents and methodologies - are the theories ofself-organization, complex systems, synergetics, dynamical systems, turbulence, catastrophes, in- stabilities, nonlinearity, stochastic processes, chaos, neural networks,cellular au- tomata, adaptive systems, andgenetic algorithms. The topics treated within Springer Complexity are as diverse as lasers or fluids in physics, machine cutting phenomenaofworkpiecesorelectric circuits withfeedbackinengineering,growthofcrystalsorpatternformationinchemistry, morphogenesisinbiology,brainfunctioninneurology,behaviorofstockexchange rates in economics, or the formation of public opinion in sociology. All these seeminglyquitedifferentkindsofstructureformationhaveanumberofimportant features and underlying structures in common.These deepstructural similarities can be exploited to transfer analytical methods and understanding from onefield to another. The Springer Complexity program therefore seeks to foster cross- fertilization between the disciplines and a dialogue between theoreticians and experimentalists foradeeperunderstandingofthegeneralstructure andbehavior ofcomplex systems. The program consists of individual books, books series such as “Springer Series in Synergetics’, “Institute ofNonlinear Science”, “Physics of Neural Net- works”, and “Understanding ComplexSystems”,aswell asvariousjournals. Springer Berlin Heidelberg New York HongKong London Milan Paris Tokyo Springer Series in Synergetics Series Editor Hermann Haken InstitutfiirTheoretischePhysik undSynergetik derUniversitatStuttgart 70550 Stuttgart,Germany and CenterforComplexSystems FloridaAtlanticUniversity BocaRaton, FL33431, USA MembersoftheEditorialBoard Ake Andersson, Stockholm, Sweden Fritz Ertl, Berlin, Germany Bernold Fiedler, Berlin, Germany Yoshiki Kuramoto, Kyoto, Japan Jiirgen Kurths, Potsdam, Germany Luigi Lugiato, Milan,Italy Jiirgen Parisi, Oldenburg, Germany Peter Schuster, Wien, Austria Frank Schweitzer, SanktAugustin, Germany Didier Sornette, Los Angeles, CA, USA, and Nice, France ManuelG.Velarde, Madrid, Spain SSSyn - AnInterdisciplinarySeries onComplexSystems The success ofthe Springer Series in Synergetics has been madepossible by the contributions ofoutstanding authors whopresented their quite often pioneering results to the science communitywell beyond the borders ofa special discipline. Indeed,interdisciplinarity is one ofthe main features ofthis series. But interdis- ciplinarity is not enough: The main goal is the search for commonfeatures of self-organizing systems in a great variety of seemingly quite different systems, or, still more precisely speaking, the search for general principles underlying the spontaneous formation ofspatial, temporal or functional structures. The topics treatedmaybeasdiverseaslasersandfluidsinphysics,patternformationinchem- istry,morphogenesis inbiology,brainfunctionsin neurologyorself-organization in a city. As is witnessed by several volumes, great attention is being paid to the pivotal interplaybetween deterministic and stochastic processes, as well as to the dialoguebetween theoreticians and experimentalists. All this has contributed to a remarkable cross-fertilization between disciplines and to a deeper understanding of complex systems. The timeliness and potential of such an approach are also mirrored —- among other indicators - by numerousinterdisciplinary workshops and conferencesall over the world. C. W. Gardiner Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Third Edition With 30 Figures G;) Springer Professor Crispin W. Gardiner D.Phil. Dr.rer.nat.(h.c.) FNZIP FAPS FRSNZ VictoriaUniversityofWellington SchoolofChemicalandPhysicalSciences P.O. Box600 Wellington,NewZealand Cataloging-in-PublicationDataappliedfor Bibliographic informationpublishedby Die Deutsche Bibliothek. Die Deutsche Bibliothekliststhispub- lication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de> ISSN 0172-7389 ISBN 3-540-20882-8 Third Edition Springer-Verlag Berlin HeidelbergNewYork ISBN3-540-61634-9 SecondEditionSpringer-Verlag Berlin HeidelbergNewYork This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks. Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, in its currentversion, and permission for use must always be obtained from Springer-Verlag. Violations areliableforprosecutionundertheGermanCopyrightLaw. Springer-VerlagisapartofSpringerScience+BusinessMedia springeronline.com ©Springer-VerlagBerlinHeidelberg1985,1997,2004 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, even intheabsenceofaspecific statement,that suchnames areexempt from therelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Coverdesign:ErichKirchner,Heidelberg Printedonacid-freepaper SPIN10965948 55/3141/ba- 543.210 Foreword ThisHandbookofStochasticMethods has becomea cornerstonein the SpringerSe- ries of Synergetics. Throughits style and the material presented, this book has be- come enormously successful, as is witnessed for instance by the numerousreprint- ings it has experienced over the past more than twenty years. Stochastic methods are of fundamentalinterest for Synergetics, which deals with self-organization in many fields of science and technology. But in addition, these methods are indis- pensable for a proper treatment of many other problems. Quite generally it may be said that the theory of stochastic processes is penetrating into more and more disciplines. One ofthe more recent developments has occurredin the theory offi- nancial markets. The authorofthis book, Crispin Gardiner, has used the need for a new edition to include a whole chapteron the numerical treatmentofstochastic dif- ferential equations. Written in Gardiner’s highly appealing style, this chapter will surely find greatinterest among bothpractitioners and theoreticians. I am sure, over many years to comethe book will find the same enthusiastic responseit has found in the past. Stuttgart Hermann Haken December 2003 Preface to the Third Edition It is now nearly twenty five years since I decided to write a book on stochastic processes, and in that time there has been significant change in this field, though more in the nature oftheir applications than in new mathematical knowledge. The mostprominentdevelopmentis the emergenceofthe field ofmathematicalfinance, ofwhich the Nobel Prize winning Black-Scholes formula forthe pricing ofoptions canbe seen as the catalyst. Theessential idea, the modelling ofan uncertain interest rate in terms ofa Wiener process, is simple, but the ramifications are enormous.I have been both pleased andsurprisedto see StochasticMethods, which I conceived as a book for scientist, become a book well-knownin the field of applications of mathematical finance. I have chosen in this third edition of Handbook ofStochastic Methods to in- clude a chapter on the numerical treatment ofstochastic differential equations, as a response to populardemand, and in recognition ofthe significant progress made in this field in the past twenty years. In spite ofthis progress, the issues in the simula- tionofstochastic differentialequations do not seem tobe very widely understood— this is unfortunate, since the correct choice of algorithm can be very important in simulations. The chapter I have addedis intended to alert anyone considering a sto- chastic simulation to the concepts involved, and to guide to available software. It is not a comprehensivetreatise on stochastic numerical analysis; forthis the reader is directed to the books ofKloeden and Platen. In fact this chapter is mainly an ex- position of the bare essentials oftheir work on the numerical solution ofstochastic differential equations. I have also deleted the former Chap. 10 on quantum Markov processes, which has now becomeobsolete. This is a fascinating field, and one in which my own interests mostly lie nowadays. It has developed very considerably since the early 1980s, and is now covered extensively in my book Quantum Noise, written with PeterZoller, and also published in the Springer Series on Synergetics. Wellington, New Zealand C.W. Gardiner November 2003 From the Preface to the First Edition Myintention in writing thisbook was to putdownin relatively simple language and in a reasonably deductive form,all those formulae and methods which have been scattered throughoutthe scientific literature on stochastic methods throughout the eighty years that they have been in use. This might seem an unnecessary aim since thereare scoresofbooksentitled “Stochastic Processes’, and similartitles, butcare- fulperusalofthese soonshowsthattheiraimdoesnotcoincide with mine. Thereare purely theoretical and highly mathematical books, there arebooksrelated to electri- cal engineeringorcommunication theory, and there arebooksforbiologists—many ofthem very good, but none ofthem covering the kind of applications that appear nowadaysso frequently in Statistical Physics, Physical Chemistry, Quantum Optics and Electronics, and ahostofother theoretical subjects that formpart ofthe subject area of Synergetics, to which series this book belongs. The main new pointofview here is the amount ofspace which deals with meth- ods of approximating problems, or transforming them for the purpose of approxi- mating them. I am fully aware that many workers will not see their methodshere. But mycriterion here has been whether an approximationis systematic. Many ap- proximationsare based on unjustifiable or uncontrollable assumptions, and arejus- tified aposteriori. Suchapproximationsarenotthe subjectofa systematic book—at least, not until they are properly formulated, and their range ofvalidity controlled. In somecases I have been able to put certain approximations on a systematic basis, and they appear here—in othercases I have not. Others have been excluded on the groundsofspace and time, and I presumethere will even be somethat have simply escaped myattention. A word on thebackground assumed. The readermusthave agoodknowledgeof practical calculus including contour integration, matrix algebra, differential equa- tions, both ordinary and partial, at the level expected of a first degree in applied mathematics, physics ortheoretical chemistry. This is not a text bookfor aparticu- larcourse, thoughit includes matter that hasbeen used in the University ofWaikato in a graduate course in physics. It contains material which I would expect any stu- dent completing a doctorate in our quantum optics and stochastic processes theory group to be familiar with. There is thus a certain bias towards my owninterests, whichis the prerogative ofan author. I expect the readership to consist mainly oftheoretical physicists and chemists, and thus the general standard is that of these people. This is not a rigorous book in the mathematical sense, but it contains results, all of which I am confident are provable rigorously, and whose proofs can be developed out ofthe demonstrations given. The organisation of the bookis as in the following table, and mightraise some eyebrows. For, after introducing the general properties ofMarkov processes, VU Preface to the First Edition 2. Probability Concepts 1. Introduction and Definitions ‘ A 3. Markov Processes eooeer eee ee ee we eo 4. Ito Calculus and A Stochastic Differential Equations 5. The Fokker-Planck A Equation ‘ 6.Approximation Methods A forDiffusion Processes Cr ey 7. MasterEquations and Jump Processes ‘ 9. Bistability, Metastability, 8. Spatially Distributed Systems and Escape Problems ‘ 10. Simulation ofStochastic Differential Equations
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