servations on the Particles of Bodies, Philosophical Magazine N. S. 8

(1830), 296.

Chapter 3

The period before Einstein

I have found no reference to a publication on Brownian motion be-

tween 1831 and 1857. Reading papers published in the sixties and sev-

enties, however, one has the feeling that awareness of the phenomenon

remained widespread (it could hardly have failed to, as it was something

of a nuisance to microscopists). Knowledge of Brown™s work reached lit-

erary circles. In George Eliot™s “Middlemarch” (Book II, Chapter V,

published in 1872) a visitor to the vicar is interested in obtaining one of

the vicar™s biological specimens and proposes a barter: “I have some sea

mice. . . . And I will throw in Robert Brown™s new thing,”˜Microscopic

Observations on the Pollen of Plants,™”if you don™t happen to have it

already.”

From the 1860s on, many scientists worked on the phenomenon. Most

of the hypotheses that were advanced could have been ruled out by con-

sideration of Brown™s experiment of the microscopic water drop enclosed

in oil. The ¬rst to express a notion close to the modern theory of Brown-

ian motion was Wiener in 1863. A little later Carbonelle claimed that the

internal movements that constitute the heat content of ¬‚uids is well able

to account for the facts. A passage emphasizing the probabilistic aspects

is quoted by Perrin [6, p. 4]:

“In the case of a surface having a certain area, the molecular col-

lisions of the liquid which cause the pressure, would not produce any

perturbation of the suspended particles, because these, as a whole, urge

the particles equally in all directions. But if the surface is of area less

than is necessary to ensure the compensation of irregularities, there is no

longer any ground for considering the mean pressure; the inequal pres-

sures, continually varying from place to place, must be recognized, as the

9

10 CHAPTER 3

law of large numbers no longer leads to uniformity; and the resultant will

not now be zero but will change continually in intensity and direction.

Further, the inequalities will become more and more apparent the smaller

the body is supposed to be, and in consequence the oscillations will at

the same time become more and more brisk . . . ”

There was no unanimity in this view. Jevons maintained that pedesis

was electrical in origin. Ord, who attributed Brownian motion largely

to “the intestine vibration of colloids”, attacked Jevons™ views [7], and I

cannot refrain from quoting him:

“I may say that before the publication of Dr. Jevons™ observations I

had made many experiments to test the in¬‚uence of acids [upon Brownian

movements], and that my conclusions entirely agree with his. In stating

this, I have no intention of derogating from the originality of of Professor

Jevons, but simply of adding my testimony to his on a matter of some

importance. . . .

“The in¬‚uence of solutions of soap upon Brownian movements, as

set forth by Professor Jevons, appears to me to support my contention

in the way of agreement. He shows that the introduction of soap in

the suspending ¬‚uid quickens and makes more persistent the movements

of the suspended particles. Soap in the eyes of Professor Jevons acts

conservatively by retaining or not conducting electricity. In my eyes it is

a colloid, keeping up movements by revolutionary perturbations. . . . It is

interesting to remember that, while soap is probably our best detergent,

boiled oatmeal is one of its best substitutes. What this may be as a

conductor of electricity I do not know, but it certainly is a colloid mixture

or solution.”

Careful experiments and arguments supporting the kinetic theory were

made by Gouy. From his work and the work of others emerged the fol-

lowing main points (cf. [6]):

1. The motion is very irregular, composed of translations and rota-

tions, and the trajectory appears to have no tangent.

2. Two particles appear to move independently, even when they ap-

proach one another to within a distance less than their diameter.

3. The motion is more active the smaller the particles.

4. The composition and density of the particles have no e¬ect.

5. The motion is more active the less viscous the ¬‚uid.

THE PERIOD BEFORE EINSTEIN 11

6. The motion is more active the higher the temperature.

7. The motion never ceases.

In discussing 1, Perrin mentions the mathematical existence of no-

where di¬erentiable curves. Point 2 had been noticed by Brown, and it is

a strong argument against gross mechanical explanations. Perrin points

out that 6 (although true) had not really been established by observation,

since for a given ¬‚uid the viscosity usually changes by a greater factor

than the absolute temperature, so that the e¬ect 5 dominates 6. Point 7

was established by observing a sample over a period of twenty years,

and by observations of liquid inclusions in quartz thousands of years old.

This point rules out all attempts to explain Brownian motion as a non-

equilibrium phenomenon.

By 1905, the kinetic theory, that Brownian motion of microscopic par-

ticles is caused by bombardment by the molecules of the ¬‚uid, seemed the

most plausible. The seven points mentioned above did not seem to be

in con¬‚ict with this theory. The kinetic theory appeared to be open to

a simple test: the law of equipartition of energy in statistical mechan-

ics implied that the kinetic energy of translation of a particle and of a

molecule should be equal. The latter was roughly known (by a determina-

tion of Avogadro™s number by other means), the mass of a particle could

be determined, so all one had to measure was the velocity of a particle

in Brownian motion. This was attempted by several experimenters, but

the result failed to con¬rm the kinetic theory as the two values of kinetic

energy di¬ered by a factor of about 100,000. The di¬culty, of course,

was point 1 above. What is meant by the velocity of a Brownian par-

ticle? This is a question that will recur throughout these lectures. The

success of Einstein™s theory of Brownian motion (1905) was largely due

to his circumventing this question.

References

[6]. Jean Perrin, Brownian movement and molecular reality, translated

from the Annales de Chimie et de Physique, 8me Series, 1909, by F. Soddy,

Taylor and Francis, London, 1910.

[7]. William M. Ord, M.D., On some Causes of Brownian Movements,

Journal of the Royal Microscopical Society, 2 (1879), 656“662.

The following also contain historical remarks (in addition to [6]). You

are advised to consult at most one account, since they contradict each

12 CHAPTER 3

other not only in interpretation but in the spelling of the names of some

of the people involved.

[8]. Jean Perrin, “Atoms”, translated by D. A. Hammick, Van Nostrand,

1916. (Chapters III and IV deal with Brownian motion, and they are sum-

marized in the author™s article Brownian Movement in the Encyclopaedia

Britannica.)

[9]. E. F. Burton, The Physical Properties of Colloidal Solutions, Long-

mans, Green and Co., London, 1916. (Chapter IV is entitled The Brow-

nian Movement. Some of the physics in this chapter is questionable.)

[10]. Albert Einstein, Investigations on the Theory of the Brownian Move-

ment, edited with notes by R. F¨rth, translated by A. D. Cowper, Dover,

u

1956. (F¨rth™s ¬rst note, pp. 86“88, is historical.)

u

[11]. R. Bowling Barnes and S. Silverman, Brownian Motion as a Natural

Limit to all Measuring Processes, Reviews of Modern Physics 6 (1934),

162“192.

Chapter 4

Albert Einstein

It is sad to realize that despite all of the hard work that had gone into

the study of Brownian motion, Einstein was unaware of the existence of

the phenomenon. He predicted it on theoretical grounds and formulated

a correct quantitative theory of it. (This was in 1905, the same year he

discovered the special theory of relativity and invented the photon.) As

he describes it [12, p. 47]:

“Not acquainted with the earlier investigations of Boltzmann and

Gibbs, which had appeared earlier and actually exhausted the subject,

I developed the statistical mechanics and the molecular-kinetic theory of

thermodynamics which was based on the former. My major aim in this

was to ¬nd facts which would guarantee as much as possible the exis-

tence of atoms of de¬nite ¬nite size. In the midst of this I discovered

that, according to atomistic theory, there would have to be a movement

of suspended microscopic particles open to observation, without know-

ing that observations concerning the Brownian motion were already long

familiar.”

By the time his ¬rst paper on the subject was written, he had heard

of Brownian motion [10, §3, p. 1]:

“It is possible that the movements to be discussed here are identical

with the so-called ˜Brownian molecular motion™; however, the information

available to me regarding the latter is so lacking in precision, that I can

form no judgment in the matter.”

There are two parts to Einstein™s argument. The ¬rst is mathematical

and will be discussed later (Chapter 5). The result is the following: Let

ρ = ρ(x, t) be the probability density that a Brownian particle is at x at

time t. Then, making certain probabilistic assumptions (some of them

13

14 CHAPTER 4

implicit), Einstein derived the di¬usion equation

‚ρ

= D∆ρ (4.1)

‚t

where D is a positive constant, called the coe¬cient of di¬usion. If the

particle is at 0 at time 0 so that ρ(x, 0) = δ(x) then

1 |x|2

’ 4Dt

ρ(x, t) = e (4.2)

(4πDt)3/2

(in three-dimensional space, where |x| is the Euclidean distance of x from

the origin).

The second part of the argument, which relates D to other physical

quantities, is physical. In essence, it runs as follows. Imagine a suspension

of many Brownian particles in a ¬‚uid, acted on by an external force K,

and in equilibrium. (The force K might be gravity, as in the ¬gure, but

the beauty of the argument is that K is entirely virtual.)

Figure 1

In equilibrium, the force K is balanced by the osmotic pressure forces

of the suspension,

grad ν

K = kT . (4.3)

ν

Here ν is the number of particles per unit volume, T is the absolute

temperature, and k is Boltzmann™s constant. Boltzmann™s constant has

ALBERT EINSTEIN 15

the dimensions of energy per degree, so that kT has the dimensions of

energy. A knowledge of k is equivalent to a knowledge of Avogadro™s

number, and hence of molecular sizes. The right hand side of (4.3) is

derived by applying to the Brownian particles the same considerations

that are applied to gas molecules in the kinetic theory.

The Brownian particles moving in the ¬‚uid experience a resistance

due to friction, and the force K imparts to each particle a velocity of the

form

K

,

mβ

where β is a constant with the dimensions of frequency (inverse time) and

m is the mass of the particle. Therefore

νK

mβ

particles pass a unit area per unit of time due to the action of the force K.

On the other hand, if di¬usion alone were acting, ν would satisfy the

di¬usion equation

‚ν

= D∆ν

‚t

so that

’D grad ν

particles pass a unit area per unit of time due to di¬usion. In dynamical

equilibrium, therefore,

νK

= D grad ν. (4.4)

mβ

Now we can eliminate K and ν between (4.3) and (4.4), giving Einstein™s

formula

kT

D= . (4.5)

mβ

This formula applies even when there is no force and when there is only

one Brownian particle (so that ν is not de¬ned).

16 CHAPTER 4

Parenthetically, if we divide both sides of (4.3) by mβ, and use (4.5),

we obtain

K grad ν

=D .

mβ ν

The probability density ρ is just the number density ν divided by the

total number of particles, so this can be rewritten as

K grad ρ

=D .

mβ ρ

Since the left hand side is the velocity acquired by a particle due to the

action of the force,

grad ρ

D (4.6)

ρ

is the velocity required of the particle to counteract osmotic e¬ects.

If the Brownian particles are spheres of radius a, then Stokes™ theory

of friction gives mβ = 6π·a, where · is the coe¬cient of viscosity of the

¬‚uid, so that in this case

kT

D= . (4.7)

6π·a

The temperature T and the coe¬cient of viscosity · can be measured,

with great labor a colloidal suspension of spherical particles of fairly uni-

form radius a can be prepared, and D can be determined by statistical

observations of Brownian motion using (4.2). In this way Boltzmann™s

constant k (or, equivalently, Avogadro™s number) can be determined. This