An Introduction to Stochastic Processes in Physics - D. Lemons.pdf

An Introduction to Stochastic
Processes in Physics
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
c 2002 The Johns Hopkins University Press
All rights reserved. Published 2002
Printed in the United States of America on acid-free paper
987654321
The Johns Hopkins University Press
2715 North Charles Street
Baltimore, Maryland 21218-4363
Library of Congress Cataloging-in-Publication Data
Lemons, Don S. (Don Stephen), 1949–
An introduction to stochastic processes in physics / by Don S. Lemons
p. cm.
Includes bibliographical references and index.
ISBN 0-8018-6866-1 (alk. paper) – ISBN 0-8018-6867-X (pbk. : alk. paper)
1. Stochastic processes. 2. Mathematical physics. I. Langevin, Paul, 1872–1946. II. Title.
L45 2001

923–dc21 2001046459
A catalog record for this book is available from the British Library
ForAllison, Nathan, and Micah
Contents
Preface and Acknowledgments xi
Chapter 1 – Random Variables 1
Random and Sure Variables 1
Assigning Probabilities 2
The Meaning of Independence 4
Problems: . Coin Flipping, . Independent Failure Modes 5
Chapter 2 – Expected Values 7
Moments 7
Mean Sum Theorem 9
Variance Sum Theorem 10
Measurements 12
Problems: . Dice Parameters, . Perfect Linear Correlation,
. Resistors in Series, . Density Fluctuations 14
Chapter 3 – Random Steps 17
Brownian Motion Described 17
Brownian Motion Modeled 18
Critique and Prospect 19
Problems: . Two-Dimensional Random Walk, . Random Walk
with Hesitation, . Multistep Walk, . Autocorrelation,
. Frequency of Heads 20
Chapter 4 – Continuous Random Variables 23
Probability Densities 23
Uniform, Normal, and Cauchy Densities 24
Moment-Generating Functio