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Dynamical Theories
of
Brownian Motion


second edition



by
Edward Nelson
Department of Mathematics
Princeton University
Copyright c 1967, by Princeton University Press.
All rights reserved.

Second edition, August 2001. Posted on the Web at
http://www.math.princeton.edu/∼nelson/books.html
Preface to the Second Edition


On July 2, 2001, I received an email from Jun Suzuki, a recent grad-
uate in theoretical physics from the University of Tokyo. It contained a
request to reprint “Dynamical Theories of Brownian Motion”, which was
¬rst published by Princeton University Press in 1967 and was now out
of print. Then came the extraordinary statement: “In our seminar, we
found misprints in the book and I typed the book as a TeX ¬le with mod-
i¬cations.” One does not receive such messages often in one™s lifetime.
So, it is thanks to Mr. Suzuki that this edition appears. I modi¬ed
his ¬le, taking the opportunity to correct my youthful English and make
minor changes in notation. But there are no substantive changes from
the ¬rst edition.
My hearty thanks also go to Princeton University Press for permis-
sion to post this volume on the Web. Together with all mathematics
books in the Annals Studies and Mathematical Notes series, it will also
be republished in book form by the Press.

Fine Hall
August 25, 2001
Contents

1. Apology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Robert Brown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3. The period before Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4. Albert Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5. Derivation of the Wiener process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6. Gaussian processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7. The Wiener integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8. A class of stochastic di¬erential equations . . . . . . . . . . . . . . . . . . . 37
9. The Ornstein-Uhlenbeck theory of Brownian motion . . . . . . . . . 45
10. Brownian motion in a force ¬eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11. Kinematics of stochastic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12. Dynamics of stochastic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
13. Kinematics of Markovian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
14. Remarks on quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
15. Brownian motion in the aether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
16. Comparison with quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 111
Chapter 1

Apology

It is customary in Fine Hall to lecture on mathematics, and any major
deviation from that custom requires a defense.
It is my intention in these lectures to focus on Brownian motion as a
natural phenomenon. I will review the theories put forward to account
for it by Einstein, Smoluchowski, Langevin, Ornstein, Uhlenbeck, and
others. It will be my conjecture that a certain portion of current physical
theory, while mathematically consistent, is physically wrong, and I will
propose an alternative theory.
Clearly, the chances of this conjecture being correct are exceedingly
small, and since the contention is not a mathematical one, what is the
justi¬cation for spending time on it? The presence of some physicists in
the audience is irrelevant. Physicists lost interest in the phenomenon of
Brownian motion about thirty or forty years ago. If a modern physicist is
interested in Brownian motion, it is because the mathematical theory of
Brownian motion has proved useful as a tool in the study of some models
of quantum ¬eld theory and in quantum statistical mechanics. I believe
that this approach has exciting possibilities, but I will not deal with it
in this course (though some of the mathematical techniques that will be
developed are relevant to these problems).
The only legitimate justi¬cation is a mathematical one. Now “applied
mathematics” contributes nothing to mathematics. On the other hand,
the sciences and technology do make vital contribution to mathematics.
The ideas in analysis that had their origin in physics are so numerous and
so central that analysis would be unrecognizable without them.
A few years ago topology was in the doldrums, and then it was re-
vitalized by the introduction of di¬erential structures. A signi¬cant role

1
2 CHAPTER 1

in this process is being played by the qualitative theory of ordinary dif-
ferential equations, a subject having its roots in science and technology.
There was opposition on the part of some topologists to this process, due
to the loss of generality and the impurity of methods.
It seems to me that the theory of stochastic processes is in the dol-
drums today. It is in the doldrums for the same reason, and the remedy
is the same. We need to introduce di¬erential structures and accept the
corresponding loss of generality and impurity of methods. I hope that a
study of dynamical theories of Brownian motion can help in this process.
Professor Rebhun has very kindly prepared a demonstration of Brown-
ian motion in Mo¬et Laboratory. This is a live telecast from a microscope.
It consists of carmine particles in acetone, which has lower viscosity than
water. The smaller particles have a diameter of about two microns (a
micron is one thousandth of a millimeter). Notice that they are more
active than the larger particles. The other sample consists of carmine
particles in water”they are considerably less active. According to the-
ory, nearby particles are supposed to move independently of each other,
and this appears to be the case.
Perhaps the most striking aspect of actual Brownian motion is the ap-
parent tendency of the particles to dance about without going anywhere.
Does this accord with theory, and how can it be formulated?
One nineteenth century worker in the ¬eld wrote that although the
terms “titubation” and “pedesis” were in use, he preferred “Brownian
movements” since everyone at once knew what was meant. (I looked up
these words [1]. Titubation is de¬ned as the “act of titubating; specif.,
a peculiar staggering gait observed in cerebellar and other nervous dis-
turbance”. The de¬nition of pedesis reads, in its entirety, “Brownian
movement”.) Unfortunately, this is no longer true, and semantical con-
fusion can result. I shall use “Brownian motion” to mean the natural
phenomenon. The common mathematical model of it will be called (with
ample historical justi¬cation) the “Wiener process”.
I plan to waste your time by considering the history of nineteenth
century work on Brownian motion in unnecessary detail. We will pick
up a few facts worth remembering when the mathematical theories are
discussed later, but only a few. Studying the development of a topic in
science can be instructive. One realizes what an essentially comic activity
scienti¬c investigation is (good as well as bad).
APOLOGY 3

Reference

[1]. Webster™s New International Dictionary, Second Edition, G. & C.
Merriam Co., Spring¬eld, Mass. (1961).
Chapter 2

Robert Brown

Robert Brown sailed in 1801 to study the plant life of the coast of Aus-
tralia. This was only a few years after a botanical expedition to Tahiti
aboard the Bounty ran into unexpected di¬culties. Brown returned to
England in 1805, however, and became a distinguished botanist. Al-
though Brown is remembered by mathematicians only as the discoverer
of Brownian motion, his biography in the Encyclopaedia Britannica makes
no mention of this discovery.
Brown did not discover Brownian motion. After all, practically anyone
looking at water through a microscope is apt to see little things moving
around. Brown himself mentions one precursor in his 1828 paper [2] and
ten more in his 1829 paper [3], starting at the beginning with Leeuwen-
hoek (1632“1723), including Bu¬on and Spallanzani (the two protago-
nists in the eighteenth century debate on spontaneous generation), and
one man (Bywater, who published in 1819) who reached the conclusion
(in Brown™s words) that “not only organic tissues, but also inorganic sub-
stances, consist of what he calls animated or irritable particles.”
The ¬rst dynamical theory of Brownian motion was that the particles
were alive. The problem was in part observational, to decide whether
a particle is an organism, but the vitalist bugaboo was mixed up in it.
Writing as late as 1917, D™Arcy Thompson [4] observes: “We cannot,
indeed, without the most careful scrutiny, decide whether the movements
of our minutest organisms are intrinsically ˜vital™ (in the sense of being
beyond a physical mechanism, or working model) or not.” Thompson
describes some motions of minute organisms, which had been ascribed to
their own activity, but which he says can be explained in terms of the
physical picture of Brownian motion as due to molecular bombardment.

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6 CHAPTER 2

On the other hand, Thompson describes an experiment by Karl Przibram,
who observed the position of a unicellular organism at ¬xed intervals. The
organism was much too active, for a body of its size, for its motion to
be attributed to molecular bombardment, but Przibram concluded that,
with a suitable choice of di¬usion coe¬cient, Einstein™s law applied!
Although vitalism is dead, Brownian motion continues to be of interest
to biologists. Some of you heard Professor Rebhun describe the problem
of disentangling the Brownian component of some unexplained particle
motions in living cells.
Some credit Brown with showing that the Brownian motion is not vital
in origin; others appear to dismiss him as a vitalist. It is of interest to
follow Brown™s own account [2] of his work. It is one of those rare papers
in which a scientist gives a lucid step-by-step account of his discovery and
reasoning.
Brown was studying the fertilization process in a species of ¬‚ower
which, I believe likely, was discovered on the Lewis and Clark expedi-
tion. Looking at the pollen in water through a microscope, he observed
small particles in “rapid oscillatory motion.” He then examined pollen
of other species, with similar results. His ¬rst hypothesis was that Brow-
nian motion was not only vital but peculiar to the male sexual cells of
plants. (This we know is not true”the carmine particles that we saw
were derived from the dried bodies of female insects that grow on cactus
plants in Mexico and Central America.) Brown describes how this view
was modi¬ed:
“In this stage of the investigation having found, as I believed, a pecu-
liar character in the motions of the particles of pollen in water, it occurred
to me to appeal to this peculiarity as a test in certain Cryptogamous
plants, namely Mosses, and the genus Equisetum, in which the existence
of sexual organs had not been universally admitted. . . . But I at the same
time observed, that on bruising the ovules or seeds of Equisetum, which at
¬rst happened accidentally, I so greatly increased the number of moving
particles, that the source of the added quantity could not be doubted. I
found also that on bruising ¬rst the ¬‚oral leaves of Mosses, and then all
other parts of those plants, that I readily obtained similar particles, not
in equal quantity indeed, but equally in motion. My supposed test of the
male organ was therefore necessarily abandoned.
“Re¬‚ecting on all the facts with which I had now become acquainted,
I was disposed to believe that the minute spherical particles or Molecules
of apparently uniform size, . . . were in reality the supposed constituent
ROBERT BROWN 7

or elementary molecules of organic bodies, ¬rst so considered by Bu¬on
and Needham . . . ”
He examined many organic substances, ¬nding the motion, and then
looked at mineralized vegetable remains: “With this view a minute por-
tion of silici¬ed wood, which exhibited the structure of Coniferae, was
bruised, and spherical particles, or molecules in all respects like those
so frequently mentioned, were readily obtained from it; in such quantity,
however, that the whole substance of the petrifaction seemed to be formed
of them. From hence I inferred that these molecules were not limited to
organic bodies, nor even to their products.”
He tested this inference on glass and minerals: “Rocks of all ages,
including those in which organic remains have never been found, yielded
the molecules in abundance. Their existence was ascertained in each of
the constituent minerals of granite, a fragment of the Sphinx being one
of the specimens observed.”
Brown™s work aroused widespread interest. We quote from a report
[5] published in 1830 of work of Muncke in Heidelberg:
“This motion certainly bears some resemblance to that observed in
infusory animals, but the latter show more of a voluntary action. The idea
of vitality is quite out of the question. On the contrary, the motions may
be viewed as of a mechanical nature, caused by the unequal temperature
of the strongly illuminated water, its evaporation, currents of air, and
heated currents, &c. ”
Of the causes of Brownian motion, Brown [3] writes:
“I have formerly stated my belief that these motions of the particles
neither arose from currents in ¬‚uid containing them, nor depended on that
intestine motion which may be supposed to accompany its evaporation.
“These causes of motion, however, either singly or combined with
other,”as, the attractions and repulsions among the particles themselves,
their unstable equilibrium in the ¬‚uid in which they are suspended, their
hygrometrical or capillary action, and in some cases the disengagement
of volatile matter, or of minute air bubbles,”have been considered by
several writers as su¬ciently accounting for the appearance.”
He refutes most of these explanations by describing an experiment in
which a drop of water of microscopic size immersed in oil, and containing
as few as one particle, exhibits the motion unabated.
Brown denies having stated that the particles are animated. His the-
ory, which he is careful never to state as a conclusion, is that matter is
composed of small particles, which he calls active molecules, which exhibit
8 CHAPTER 2

a rapid, irregular motion having its origin in the particles themselves and
not in the surrounding ¬‚uid.
His contribution was to establish Brownian motion as an important
phenomenon, to demonstrate clearly its presence in inorganic as well as
organic matter, and to refute by experiment facile mechanical explana-
tions of the phenomenon.

References

[2]. Robert Brown, A brief Account of Microscopical Observations made
in the Months of June, July, and August, 1827, on the Particles contained
in the Pollen of Plants; and on the general Existence of active Molecules
in Organic and Inorganic Bodies, Philosophical Magazine N. S. 4 (1828),
161“173.
[3]. Robert Brown, Additional Remarks on Active Molecules, Philosophi-
cal Magazine N. S. 6 (1829), 161“166.
[4]. D™Arcy W. Thompson, “Growth and Form”, Cambridge University
Press (1917).

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