An introduction to stochastic processes in physics, containing On the theory of Brownian notion

124 Pages·2002·0.398 MB·English

by  Don S. Lemons

AnIntroductiontoStochastic
ProcessesinPhysics

An Introduction to
Stochastic Processes
in Physics
Containing “On the Theory of Brownian
Motion” by Paul Langevin, translated by
Anthony Gythiel
DON S. LEMONS
THE JOHNS HOPKINS UNIVERSITY PRESS
BALTIMORE AND LONDON
Copyright(cid:1)c 2002TheJohnsHopkinsUniversityPress
Allrightsreserved.Published2002
PrintedintheUnitedStatesofAmericaonacid-freepaper
987654321
TheJohnsHopkinsUniversityPress
2715NorthCharlesStreet
Baltimore,Maryland21218-4363
www.press.jhu.edu
LibraryofCongressCataloging-in-PublicationData
Lemons,DonS.(DonStephen),1949–
Anintroductiontostochasticprocessesinphysics/byDonS.Lemons
p.cm.
Includesbibliographicalreferencesandindex.
ISBN0-8018-6866-1(alk.paper)–ISBN0-8018-6867-X(pbk.:alk.paper)
1.Stochasticprocesses.2.Mathematicalphysics.I.Langevin,Paul,1872–1946.II.Title.
QC20.7.S8L452001
530.15(cid:2)923–dc21 2001046459
AcatalogrecordforthisbookisavailablefromtheBritishLibrary
ForAllison,Nathan,andMicah

Contents
PrefaceandAcknowledgments xi
Chapter1 – RandomVariables 1
1.1 RandomandSureVariables 1
1.2 AssigningProbabilities 2
1.3 TheMeaningofIndependence 4
Problems: 1.1.CoinFlipping,1.2.IndependentFailureModes 5
Chapter2 – ExpectedValues 7
2.1 Moments 7
2.2 MeanSumTheorem 9
2.3 VarianceSumTheorem 10
2.4 CombiningMeasurements 12
Problems: 2.1.DiceParameters,2.2.PerfectLinearCorrelation,
2.3.ResistorsinSeries,2.4.DensityFluctuations 14
Chapter3 – RandomSteps 17
3.1 BrownianMotionDescribed 17
3.2 BrownianMotionModeled 18
3.3 CritiqueandProspect 19
Problems: 3.1.Two-DimensionalRandomWalk, 3.2.RandomWalk
withHesitation,3.3.MultistepWalk,3.4.Autocorrelation,
3.5.FrequencyofHeads 20
Chapter4 – ContinuousRandomVariables 23
4.1 ProbabilityDensities 23
4.2 Uniform,Normal,andCauchyDensities 24
4.3 Moment-GeneratingFunctions 27
Problems: 4.1.Single-SlitDiffraction,4.2.MomentsofaNormal,
4.3.ExponentialRandomVariable,4.4.PoissonRandomVariable 29
Chapter5 – NormalVariableTheorems 33
5.1 NormalLinearTransformTheorem 33
5.2 NormalSumTheorem 34
5.3 JointlyNormalVariables 35
viii CONTENTS
5.4 CentralLimitTheorem 36
Problems: 5.1.UniformLinearTransform,5.2.AddingUniform
Variables,5.3.DependentNormals 39
Chapter6 – Einstein’sBrownianMotion 41
6.1 SureProcesses 41
6.2 WienerProcess 43
6.3 BrownianMotionRevisited 45
6.4 MonteCarloSimulation 46
6.5 DiffusionEquation 48
Problems: 6.1. Autocorrelated Process, 6.2. Concentration Pulse,
6.3. Brownian Motion with Drift, 6.4. Brownian Motion in a
Plane 49
Chapter7 – Ornstein-UhlenbeckProcesses 53
7.1 LangevinEquation 53
7.2 SolvingtheLangevinEquation 54
7.3 SimulatingtheO-UProcess 57
7.4 Fluctuation-DissipationTheorem 59
7.5 JohnsonNoise 60
Problems: 7.1.TerminalSpeed,7.2.RLCircuit 62
Chapter8 – Langevin’sBrownianMotion 63
8.1 IntegratingtheO-UProcess 63
8.2 SimulatingLangevin’sBrownianMotion 66
8.3 SmoluchowskiApproximation 68
8.4 Example: BrownianProjectile 69
Problems:8.1.Derivation,8.2.X-VCorrelation,8.3.RangeVariation 72
Chapter9 – OtherPhysicalProcesses 75
9.1 StochasticDampedHarmonicOscillator 75
9.2 StochasticCyclotronMotion 80
Problems: 9.1.SmoluchowskiOscillator,9.2.CriticalDamping,
9.3.OscillatorEnergy,9.4.O-UProcessLimit,9.5.Statistical
Independence 83
Chapter10 – FluctuationswithoutDissipation 85
10.1 Effusion 85
10.2 ElasticScattering 88
Problems:10.1.Two-LevelAtoms,10.2.Cross-FieldDiffusion,10.3.Mean
SquareDisplacement 94
CONTENTS ix
Appendix A: “On the Theory of Brownian Motion,” by Paul
Langevin,translatedbyAnthonyGythiel 97
AppendixB:KineticEquations 101
AnswerstoProblems 103
References 107
Index 109