Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences

440 Pages·2004·25.49 MB·English

by  Gardiner, Crispin W.

SPRINGER ass erleas
IN SYNERGETICS COMPLE
C. W. Gardiner
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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
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ISSN 0172-7389
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ISBN3-540-61634-9 SecondEditionSpringer-Verlag Berlin HeidelbergNewYork
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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