Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability: Volume 3 Contributions to Astronomy, Meteorology, and Physics [Reprint 2020 ed.] 9780520323438

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PROCEEDINGS OF FOURTH

BERKELEY

VOLUME I I I

THE

SYMPOSIUM

PROCEEDINGS of the FOURTH BERKELEY SYMPOSIUM ON MATHEMATICAL STATISTICS AND PROBABILITY Held at the Statistical Laboratory University of California June 20—July 30, 1960, with the support of

University of California National Science Foundation Office of Naval Research Office of Ordnance Research Air Force Office of Research National Institutes of Health

V O L U M E III CONTRIBUTIONS TO ASTRONOMY,

METEOROLOGY,

EDITED BY J E R Z Y

UNIVERSITY

OF

NEYMAN

CALIFORNIA

BERKELEY AND LOS ANGELES

1961

AND

PRESS

PHYSICS

UNIVERSITY OF CALIFORNIA

PRESS

B E R K E L E Y AND LOS A N G E L E S CALIFORNIA

CAMBRIDGE U N I V E R S I T Y LONDON,

PRESS

ENGLAND

© 1961,

BY

THE REGENTS OF THE UNIVERSITY OF

CALIFORNIA

The United States Government and its offices, agents, a n i employees, acting within the scope of their duties, may reproduce, publish, and use this material in whole or in part for governmental purposes without payment of royalties thereon or therefor. The publication or republication by the government either separately or in a public document of any material in which copyright subsists shall not be taken to cause any abridgment or annulment of the copyright or to authorize any use or appropriation of such copyright material without the consent of the copyright proprietor. LIBRARY O F CONGRESS CATALOG CARD N U M B E R :

P R I N T E D IN T H E U N I T E D S T A T E S O F

49-8189

AMERICA

CONTENTS OF PROCEEDINGS, VOLUMES I, II, AND IV Volume I—Theory of Statistics F. J. ANSCOMBE, Examination of residuals. RICHARD BELLMAN, A mathematical formulation of variational processes of adaptive type. Z. W. BIRNBAUM, On the probabilistic theory of complex structures. DAVID BLACKWELL, Exponential error bounds for finite state channels. L. BREIMAN, Optimal gambling systems for favorable games. HERMAN CHERNOFF, Sequential tests for the mean of a normal distribution. Y. S. CHOW and H E R B E R T ROBBINS, A martingale system theorem and applications. D. R. COX, Tests of separate families of hypotheses. TORE DALENIUS, JAROSLAV H A j E K , and STEFAN ZUBRZYCKI, On plane sampling and related geometrical problems. H. E. DANIELS, The asymptotic efficiency of a maximum likelihood estimator. GEORGE B. DANTZIG and ALBERT MADANSKY, On the solution of two-stage linear programs under uncertainty. F. N. DAVID and EVELYN FIX, Rank correlation and regression in a nonnormal surface. BRUNO D E FINETTI, The Bayesian approach to the rejection of outliers. R. L. DOBRUSHIN, Mathematical problems in the Shannon theory of optimal coding of information. THOMAS S. FERGUSON, On the rejection of outliers. R. FORTET, Hypothesis testing and estimation for Laplacian functions. J. L. HODGES, Jr. and E. L. LEHMANN, Comparison of the normal scores and Wilcoxon tests. HAROLD HOTELLING, The behavior of some standard statistical tests under nonstandard conditions. W. JAMES and CHARLES STEIN, Estimation with quadratic loss. J. KIEFER, Optimum experimental designs V, with applications to systematic and rotatable designs. TOSIO KITAGAWA, Successive processes of statistical optimizing procedures. WILLIAM KRUSKAL, The coordinate-free approach to Gauss-Markov estimation, and its application to missing and extra observations. D. V. LINDLEY, The use of prior probability distributions in statistical inference and decisions. EMANUEL PARZEN, Regression analysis of continuous parameter time series. ROY RADNER, The evaluation of information in organizations. C. RADHAKRISHNA RAO, Asymptotic efficiency and limiting information. ALFRED RENYI, On measures of entropy and information. HERMAN RUBIN, The estimation of discontinuities in multivariate densities, and related problems in stochastic processes. LEONARD J. SAVAGE, The foundations of statistics reconsidered. L. SCHMETTERER, Stochastic approximation. CLAUDE E. SHANNON, Two-way communication channels. H E R B E R T SOLOMON, On the distribution of quadratic forms in normal variates. ANTONlN SPACER, Statistical estimation of semantic provability. A. J. THOMASIAN, The metric structure of codes for the binary symmetric channel. JOHN W. TUKEY, Curves as parameters, and touch estimation. I. VINCZE, On two-sample tests based on order statistics. S. S. WILKS, A combinatorial test for the problem of two samples from continuous distributions. HERMAN O. A. WOLD, Unbiased predictors. J. WOLFOWITZ, A channel with infinite memory.

Volume II—Probability Theory B. V. GNEDENKO, Alexander Iacovlevich Khinchin. J. L. DOOB, Appreciation of Khinchin. YU. K. BELAYEV, Continuity and Holder's conditions for sample functions of stationary Gaussian Processes. K. L. CHUNG, Probabilistic methods in Markov chains. HARALD CRAMlCR, On some classes of nonstationary stochastic processes. J. H. CURTISS, A stochastic treatment of some classical interpolation problems. J. L. DOOB, Notes on martingale theory. A. DVORETZKY, P. ERDOS, and S. KAKUTANI, Nonincrease everyv

VI

CONTENTS OF

PROCEEDINGS

where of the Brownian motion process. E. B. DYNKIN, Transformations of Markov processes connected with additive functionals. MAREK FISZ, Characterization of sample functions of stochastic processes by some absolute probabilities. B. V. GNEDENKO, V. S. KOROLUR, and A. V. SKOROKHOD, Asymptotic expansions in probability theory. ULF GRENANDER, Stochastic groups and related structures. OTTO HAN§, Random operator equations. HENRY HELSON and DAVID LOWDENSLAGER, Vector-valued processes. WASSILY HOEFFDING,On sequences of sums of independent random vectors. KIYOSI ITO,Wiener integral and Feynman integral. SHIZUO KAKUTANI, Spectral analysis of stationary Gaussian processes. SAMUEL KARLIN and JAMES MCGREGOR, Occupation time laws for birth and death processes. PAUL LEVY, An extension of the Lebesgue measure of linear sets. YU. V. LINNIK, On the probability of large deviations for the sums of independent variables. EUGENE LUKACS, Recent developments in the theory of characteristic functions. J. G. MAULDON, Asymmetric oriented percolation on a plane. JACQUES NEVEU, Lattice methods and submarkovian processes. E. J. G. PITMAN, Some theorems on characteristic functions of probability distributions. YU. V. PROHOROV, The method of characteristic functionals. G. E. H. REUTER, Competition processes. M. ROSENBLATT, Independence and dependence. YU. A. ROZANOV, An application of the central limit theorem. CZESLAW RYLLNARDZEWSKI, Remarks on processes of calls. WALTER L. SMITH, On some general renewal theorems for nonidentically distributed variables. FRANK SPITZER, Recurrent random walk and logarithmic potential. LAJOS TAKÁCS, The transient behavior of a single server queueing process with a Poisson input. K. URBANIK, Generalized stochastic processes with independent values. DAVID M. G. WISHART, An application of ergodic theorems in the theory of queues. A. M. YAGLOM, Second-order homogeneous random fields. KOSAKU YOSIDA, On a class of infinitesimal generators and the integration problems of evolution equations.

Volume IV—Biology and Problems of Health NIELS ARLEY, Theoretical analysis of carcinogenesis. P. ARMITAGE and R. DOLL, Stochastic models for carcinogenesis. M. S. BARTLETT, Monte Carlo studies in ecology and epidemiology. R. E. BELLMAN, J. A. JACQUEZ, and R. KALABA, Mathematical models of chemotherapy. AGNES BERGER and RUTH Z. GOLD, On comparing survival times. JOSEPH BERKSON and J. L. HODGES, Jr., A minimax estimator for the logistic function. MONES BERMAN, Application of differential equations to the study of the thyroid system. HAROLD F. BLUM, Comparable models for carcinogenesis by ultraviolet light and by chemical agents. W. RAY BRYAN, Virus carcinogenesis. D. G. CHAPMAN, Statistical problems in dynamics of exploited fisheries populations. CHIN LONG CHIANG, On the probability of death from specific causes in the presence of competing risks. GEORGE B. DANTZIG, JAMES C. DEHAVEN, and CRAWFORD F. SAMS, A mathematical model of the chemistry of the external respiratory system. W. J. DIXON, Some statistical uses of large computers. JOHN E. DUNN, Jr., Relation of carcinoma in situ to invasive carcinoma of the cervix uteri. MURRAY EDEN, A two-dimensional growth process. SEYMOUR GEISSER, The Latin square as a repeated measurements design. JOHN T. GENTRY and ELIZABETH PARKHURST, Low level radiation effects. SAMUEL W. GREENHOUSE, A stochastic process arising in the study of muscular contraction. HARDIN B. JONES, Mechanism of aging suggested from study of altered death risks. NATHAN MANTEL, Principles of chemotherapeutic screening. L. MARTIN, Stochastic processes in physiology. G. RASCH, On general laws and the meaning of measurement in psychology. JOHN L. STEPHENSON, Integral equation description of transport phenomena in biological systems. WILLIAM F. TAYLOR, On the methodology of studying aging in humans. CORNELIUS A. TOBIAS, Quantitative approaches to the cell division process. HOWARD G. TUCKER, A stochastic model for a two-stage theory of carcinogenesis. W. A. O'N. WAUGH, Age-dependence in a stochastic model of carcinogenesis.

PREFACE T H E F O U R T H B E R K E L E Y SYMPOSIUM on Mathematical Statistics and Probability was organized by the personnel of the Statistical Laboratory and of the Department of Statistics, University of California, Berkeley, comprised of Professors E. W. Barankin, D. Blackwell, F. Cogburn, E. Fix, J. L. Hodges, Jr., L. Le Cam, E. L. Lehmann, M. Lofeve, J. Neyman, R. Radner, H. SchefK, E. L. Scott, and A. J. Thomasian. The plan of the Symposium was drawn and the contributors selected with an active participation of two Advisory Committees. For advice regarding theory of probability and statistics and also regarding some applications, we are indebted to the Advisory Committee composed of delegates of the American Mathematical Society, Professor J. H. Curtiss, J. L. Doob, and William Feller, of delegates of the Institute of Mathematical Statistics, Professors Albert H. Bowker and H. E. Robbins, and of the Editor of the Annals of Mathematical Statistics, Professor W. H. Kruskal. The program on biology and on problems of health was arranged with the help of another Advisory Committee composed of representatives of the National Institutes of Health, Drs. M. Berman, R. Bryan, H. F. Dorn, S. Geisser, S. Greenhouse, J. Hearon, M. Shimkin, and J. L. Stephenson, and also Drs. A. Berger, J. Jacquez, H. B. Jones, and C. Tobias. This help is very gratefully acknowledged.

The purpose of the Berkeley Symposia is to stimulate research through the lectures of the carefully selected speakers and by providing opportunity for personal contacts extending over several weeks spent in Berkeley for scholars from different centers, and by publishing the Proceedings. The Proceedings are intended to represent a comprehensive cross section of contemporary thinking on problems of probability and mathematical statistics. Although completeness is difficult to achieve, the Statistical Laboratory is gratified by the gradual increase in the number of intellectual centers throughout the world represented at the successive Symposia. In particular, the present Proceedings are much richer than those of the earlier Symposia because of the several contributions from members of the great Russian school of probability. These contributions were secured through the kind cooperation of the National Academy of Sciences in Washington, D. C. and of the Academy of Sciences of the U.S.S.R. in Moscow, and hearty thanks are due to both institutions. Volume I of the Proceedings is given to the theory of statistics. Volume II contains papers on probability. As a preliminary section this volume contain the record of a special meeting of the Symposium dedicated to the memory of the recently deceased remarkable probabilist A. I. Khinchin. It is a pleasure to thank Professor B. V. Gnedenko for providing us with a photograph of the late Professor Khinchin. It appears as the frontispiece of Volume II. vii

viii

PREFACE

Although a comprehensive representation of theoretical developments is difficult to achieve, the field of application of the theory of probability and of statistics is currently so broad and contains so many different domains that any significant approach to completeness of coverage is simply impossible, even in a meeting extending over six weeks. Particular domains of application require separate symposia. The several domains that were actually discussed at the Fourth Symposium were selected partly on the basis of personal preference of certain scholars invited to the Symposium for the excellence of their work in a number of fields and partly because of the related research conducted at the Statistical Laboratory. Applications to physical sciences are published in Volume III of the Proceedings. They include astronomy, meteorology, and physics. Volume IV of the Proceedings is given to biology and problems of medicine. Here, with reference to one particular subject, namely theory of carcinogenesis, substantial effort was expended to collect not only a comprehensive representation of the current theoretical statistical and probabilistic work, but also papers characterizing various relevant empirical findings. Thus, the fourth volume of these Proceedings includes papers outlining the important biological studies relevant to the theory of carcinogenesis, from experiments on cellular phenomena, through studies of viruses, to large-scale surveys on the possible effects of low level radiation, one of the most dreaded carcinogens. A realistic and precise theory of carcinogenesis seems to belong to the category of those problems in which the solution depends on a close cooperation and mutual understanding of experimenters on the one hand and of statisticians on the other. It is hoped that Volume IV of these Proceedings will contribute to the establishment of such understanding and cooperation. The sources of financial support of the Fourth Symposium are listed on the title page of each volume of these Proceedings and this support is here very gratefully acknowledged. First and foremost, hearty thanks are due to Professor Clark Kerr, President of the University of California, for his generous grant made several years before the Symposium. This initial grant insured the organization of the Symposium. However, without the subsequent generous financial support of the National Science Foundation, followed by those of the Air Force Office of Research and Development, of the Office of Naval Research, and of the Office of Ordnance Research, the meeting could not have been held on the scale that actually was achieved. A special grant and also considerable moral support of the National Institutes of Health helped to organize a series of sessions on biology and medicine. In addition to direct grants of funds, the organization of the Symposium obtained a most valuable indirect support kindly provided by the U. S. Air Force Office of Scientific Research and, to an even greater extent, by the Office of Naval Research. This indirect support consisted in the air transportation for a number of foreign guests over the oceans, both the Atlantic and the Pacific.

PREFACE

ix

Without this form of help, foreign participation in the Fourth Symposium would have been greatly reduced. The growth of the Symposia is naturally accompanied by the corresponding growth of the Proceedings, from about 500 pages for the First Symposium of 1945/46 to about 2000 printed pages for the Fourth. Speedy publication of this amount of scientific material naturally presents a number of problems. This is particularly true when a substantial part of the material is originally written in foreign languages and requires translation into English. This is even more particularly true when it is desired to produce books at a relatively low price which will make them accessible to young scholars. In the above connection I am greatly indebted to a number of colleagues and friends for their work on the translation of manuscripts and for their help to simplify some of the formulas so as to make them less expensive to set in type. In particular my thanks are due to Professor Evelyn Fix, to Professor Lucien Le Cam, to Dr. Emma Lehmer, to Professor and Mrs. J. G. Mauldon, and to Professor Elizabeth L. Scott. Further thanks for help in translations are also due to Drs. I. J. Abrams, A. II. Kraiman, and L. Neustadt of the Space Technology Laboratories. For work on the preparation of some of the manuscripts for the printers my thanks go to several Research Assistants in the Statistical Laboratory, S. Bhuchongkul, N. L. Cook, M. Darland, J. Denny, J. Fabius, S. S. Jogdeo, J. Karush, W. Klonecki, J. Kraft, W. Lawton, J. L. Lovasich, P. Mikulski, L. Regelson, and G. D. Woodard. Before being sent to the printers, many manuscripts had to be retyped. This was excellently done by Mrs. Sharlee Guise and Mrs. Julia Rubalcava. Special thanks are due W. H. Newkirk who, as copy editor of the Proceedings, took care of the many complexities of publishing with uncommon care and unceasing patience. Financial difficulties connected with the publication were overcome with the effective help of the Editorial Committee of the University of California, due to the generous subvention of the IBM Company and to additional grants from the RAND Corporation and from the Space Technology Research Laboratories. All this help is gratefully acknowledged. Funds and adequate personnel are, of course, indispensable in any substantial enterprise. However, in addition to these most necessary elements, for an organization to run smoothly it is necessary that it include someone whose continuous care, initiative, foresight, and good common sense would make the organization "click." For this role in the organization of the Symposium it is a pleasure to thank my colleague and friend, Professor Elizabeth L. Scott. Mr. August Frug6, the Director of the University of California Press, was most cooperative and helpful in organizing the publication of the Proceedings. The speed with which the Proceedings are published is very essential, and one year taken to produce about 2000 pages in print is an excellent record. However,

X

PREFACE

the actual speed in manufacturing the books is even better than it looks, for, to this editor's deep shame and regret, the manuscripts of which he himself is a co-author were the last to be delivered to the Press. In fact, they were delivered in the last days of June 1961, and my thanks to Mr. Frugé are accompanied by sincere apologies. JERZY NEYMAN

Director, Statistical

Laboratory

CONTENTS J. A. CRAWFORD—The Motion of Charged Particles in a Random Magnetic Field

1

W. B. FRETTER—Problems in the Measurement of Ionization in Tracks in a Cloud Chamber

11

M. HAMMERSLEY—On the Statistical Loss of Long-Period Comets from the Solar System, I I

17

M. HAMMERSLEY—On the Dynamical Disequilibrium of Individual Particles

79

G. K E N D A L L — T h e Distribution of Energy Perturbations for Halley's and Some Other Comets

87

DAVID

G. K E N D A L L — S o m e Problems in the Theory of Comets, I

99

DAVID

G. K E N D A L L — S o m e Problems in the Theory of Comets, I I

121

J. J.

DAVID

R. H. KERR—Perturbations of Cometary Orbits L. LE CAM—A

149

Stochastic Description of Precipitation . . . .

165

J . L . LOVASICH, N . U . MAYALL, J . NEYMAN, a n d E . L . SCOTT—

The Expansion of Clusters of Galaxies

187

R. A. LYTTLETON—On the Statistical Loss of Long-Period Comets from the Solar System, I

229

R. M I N K O W S K I — T h e Luminosity Function of Extragalactic Radio Sources

245

and ELIZABETH L. S C O T T — F i e l d Galaxies: Luminosity, Redshift, and Abundance of Types. Part I. Theory . .

261

JERZY NEYMAN

THORNTON P A G E — A v e r a g e

Masses of the Double Galaxies

LAURENT S C H W A R T Z — D e n s i t y

.

.

277

of Probability of Presence of Ele-

mentary Particles

307

S. M. ULAM—On Some Statistical Properties of Dynamical Systems

315

W. J. YOUDEN—Statistical Problems Arising in the Establishment of Physical Standards

321

xi

THE MOTION OF CHARGED PARTICLES IN A RANDOM MAGNETIC FIELD J. A. CRAWFORD UNIVERSITY OF CALIFORNIA,

BERKELEY

1. Introduction There are in nature several instances of charged particles with very high energies, distributed over a wide range: cosmic rays, the electrons of the outer Van Allen belt, and protons emitted by the sun in association with flares. The origin of these particles has long presented a puzzle. In 1949, Enrico Fermi pointed out that the motion of charged particles in a randomly changing magnetic field ought to lead, through Faraday's law of induction, to a gradual but unlimited increase in their mean energy [1], Fermi proposed this mechanism as the source of cosmic rays, the random magnetic field being identified with the field expected to exist in interstellar space, according to Alfv&i's ideas on cosmical electrodynamics [2], If one wishes to study quantitatively the problem of the motion of charged particles in a random magnetic field, one is faced with the obvious difficulty that the equations of motion are complicated and cannot be integrated. I t is therefore necessary to introduce certain simplifying assumptions. We shall begin right away by listing the principal assumptions made in this paper, assumptions which are nearly the same as those laid down by Fermi in his treatment of the cosmic ray problem. 2. Outline of the problem The motion of a particle of charge q and momentum p in a magnetic field B is associated with a characteristic length I = 2wpc/\q\B, which is the distance traveled during one cyclotron period (c is the speed of light). We shall adopt three postulates concerning the nature of the random magnetic field and the charged particles moving in it. POSTULATE 1. If L is a characteristic length, suitably defined, associated with the fluctuations of the magnetic field, then l/L ) -7±.

The justification of postulate 3, which underlies the linearization procedure, results from the following considerations. In a frame of reference moving with the velocity U, the quantity p\/B is known [3], with certain restrictions, to be an adiabatic invariant of the particle motion to all orders in the small quantity l/L. Even this restricted invariance is not exact, however, as the corresponding power series in l/L does not converge to a limit. Nevertheless, one has the impression that p\/B is effectively a "good" constant and we shall assume that this is the case. Thus, if p±/mc U. To obtain v± in a stationary frame of — reference we merely have to * add to v± the normal plasma velocity component Ux. Since U/c « 1, the inequality vx/c •g r a d -> b )

( 6

'

' V±~

With the help of the usual relativistic relations between energy, momentum, and velocity, this may be rewritten (15)

d

jt log p = ( b • grad b ) • v±.

To facilitate the integration of (15) we may now utilize postulate 1. The time taken for the particle to traverse the distance I is one cyclotron period T = l/v. During this time, the increment of log p, according to (15), is of the order of —>

—•

magnitude (v ± /v)(l/L), which is negligible; furthermore b • grad b is essentially constant over the distance I. If we now proceed to average (15) over one cyclotron period, may —> we— * therefore neglect not only the change in log p but also the change in b • grad b. The averaged equation (15) may therefore be written (16)

J log p = ( b • grad b) • v±,

where

(17)

— 1 fT/2 t>AQ = r 1 J-T/2

_ drv±(t+r).

Now it may be shown [4], with the help of postulate 1, that v± is actually in-

MOTION OF CHARGED PARTICLES

5

dependent of the particle velocity and is equal to the so-called electric drift —>

—»•

velocity cE X B/B2, which, according to (1), is simply EXB U± = c ; W

(18) Thus (16) becomes (19)

d ^ l o g p = (Ò • g r a d ò ) •

X-

However, since b • grad ò is orthogonal to B, we may write this simply (20)

^ l o g p = ( ò - g r a d ò ) • £/.

Thus the right side of (20) depends, according to (18), only on the electromagnetic field and its gradient at the position of the particle. By using the relativistic relation (6), and writing v = ds/dt, we may rewrite (20) in the form (21)

| =

( 6 - gradò)

-^K(7,t),

where 6 = tanh" 1 c

(22)

and s is the distance traveled by the particle. 4. The Fokker-Planck equation To construct the Fokker-Planck equation for equation (21) we must compute the mean value limits [5], [6] fi = lim (23)

A6 = 6 - 6>„ AS

i r

j2 = lim ——> as—>o As

As = s — s0.

Here, the limit As —»0 is to be understood with the qualification As/L » 1. We shall assume that U/v is sufficiently small that (24)

(U/v)(As/L)

« 1.

This condition is somewhat stronger than is needed to satisfy the requirement that (AS2)1'2 be small. However, it insures also that the time elapsed, ~As/v, is small compared with the characteristic time L/U with which the fields B and U fluctuate. Thus the time dependence of K(r, t) may be neglected. Furthermore, the particle trajectory which, in the linearized theory follows the equation

6

FOURTH B E R K E L E Y

(25)

SYMPOSIUM:

d r =

CRAWFORD

± b d s ,

is determined entirely by the particular realization of the magnetic field and the value of r at the instant t0. In fact the trajectory lies on the magnetic "line of force" going through r0. Thus, when we integrate (21), the increment (26)

A0 =

[ * J so

* d 8 K ( r ,

+

t),

t ~ t o ,

depends only upon n, t0, and As [and on the sign in (25)]. If we now assume that B and U are stationary random functions [7] in both r and t, and average over r0, t0, and the sign, it follows that /i, fi are constants. Neglecting small quantities of order As2, (AO ), the probability distribution P(6, s) of the kinematic variable 6 then satisfies the Fokker-Planck equation 3

s!£

< m y

I,

=

ds

1

2

dP

J 2

J l

de

We shall now prove that /i vanishes. 5. Calculation of fi According to (26) we have (28)

(A0) = < / s ; + A S C / S X ( M o ) ) -

From ergodic theory we may assume that (As) is equal to the limit of the space average of the random function A0(ro,

K ( r ,

(0)

tB)

We need therefore only calculate (K). Because of the presence of irreversible processes, such as viscous dissipation, it is not obvious that (K) must vanish in a time-stationary situation, and to prove that it does we must appeal to the fundamental equation of magnetohydrodynamics,

MOTION OF CHARGED PARTICLES

7

This equation is a direct consequence of (1) and of the Maxwell-Faraday equation (31)

- ~ ^ = c ot

Cnr\E. —>

W i t h the help of the Maxwell equation div B = 0, (30) m a y be written dR m

(32)

= B -»

'

g r a d U_±

> _B > d i v_ U > x _ > ~ ~ Ul- '

grad B

-

_

Substituting (33)

grad B = (grad b)B + (grad B%,

this becomes (34)

^

= B • grad UL - B div U± -

Ux • (grad b)B -

UL • (grad B)b.

—>

Taking the scalar product of this equation with B and dividing by B2, we get (35)

|log£ = b • (grad U±) • b - div Ux —* —>

Ux- (grad b) • b - U

±

• grad log B.

Noting t h a t (grad b) • b vanishes and t h a t (36)

(grad Ux) - b=

- ( g r a d b) • U±>

we obtain (37)

£ log B + U± • grad log B + div UL = ot

-cK.

Noting t h a t (38)

|

=

grad

represents time differentiation following a point moving with the velocity we obtain (39)

jt log B + div UL =

Ux,

-cK.

I t is of interest to note [8] t h a t — cK is the local rate a t which a magnetic line of force stretches, each point on the line being assumed to move with the velocity U±. —>

—»

The expectation value of div U x vanishes, since U L is a stationary random —• function of r. Therefore

8

FOURTH B E R K E L E Y SYMPOSIUM:

(40)

(jt\oEBy

=

CRAWFORD

(-cK).

Here ( — cK) must be constant, since K is a stationary random function in space and time. If |(log B)\ is not ultimately to become infinite, this constant must be zero. The average (log B) is taken over a random sample of points moving with the velocity Ux. If, for instance on physical grounds, we impose upon log B an upper bound log£ 0 , then unless ( — cK) vanishes, (log B) must at some time exceed log Bo, a contradiction. Thus we must have (41)

(K) = 0

and therefore (Ad) and fi must vanish. Writing / 2 = 2 0, let T(x, co) be the supremum of all T' such that x + X(t, 0 whenever 0 ^ t ^ T'. In all that follows we shall tacitly suppose that X(t, co) is a continuous function of t and that T(x, co) is finite; this is justified inasmuch as the set of co, for which these suppositions are invalid, has zero ¿¿-measure. For z ^ 0, let V(z) be any nonnegative Borel-measurable function. We work throughout with a real number system extended to include the value so that V(z) and other subsequent quantities may be . We shall prove that (2.2)

G(x, u) = fQ™

V[x + X(t, »)] dt

exists as a (possibly infinite) random variable, that is, that it is ¿¿-measurable. We shall study its moment-generating function (2.3)

{x,u) = I e-«