¯ ¯

2

vx : What will be the value of vx ? This ensemble average will not be zero since vx is

2 2

either positive or zero ” never negative. I already said that molecules of water are not at rest.

They are perpetually moving in di¬erent directions, some faster than others. Now, heat is a

form of energy. When we heat water, we give energy to it. As a result, the water molecules

start moving faster. Their average kinetic energy rises. On the other hand, when we heat

water, its temperature also rises. Thus temperature is a measure of the average kinetic

energy of the water molecules. It turns out that when the suspended particle is in thermal

equilibrium with the water, its average kinetic energy is proportional to the temperature:

121

mv = kT,

2x2

where m is the mass of the suspended particle, k is a constant and T is the absolute temper-

ature of the water. Hence vx = kT /m. This implies that heavier particles will have smaller

2

vx . Similar statements can be made about the motion in y and z directions.4

2

x2 : How far from the origin will the particle be after some time? Equivalently, what

will be the value of x2 ? Before I answer this question, note the important complication in

the present problem. Now, not only the direction but also the size of each step is a variable

and is completely independent of the preceding step. (Why? Remember the ¬‚uctuating

force mentioned in section III.)

Using the ideas from Statistical Mechanics, Einstein derived the following result:

kT

x2 = t,

3π·a

where · is the viscosity of the liquid, a is the radius of the suspended particle (assumed to

be spherical) and t is the elapsed time. Thus the mean square displacement x2 increases

linearly with time (i.e., the power of t in the above equation is unity).

Looking at the last equation, can you now answer the question no. (3) in the Quiz above?

Please ¬nd out yourselves, before you read the answers given here:

8

(a) The Brownian Motion is less vigorous in cold water than in hot water. (b) The

Brownian Motion will be damped if water is replaced by a more viscous liquid. (c) A bigger

particle will drift less than a smaller particle ” we do not notice the Brownian Motion of

¬sh, people or boats. Do we?

Using Einstein™s result, one can also answer quantitatively the more speci¬c questions

listed at the beginning of section V.

VI. Importance

• In 1908, the French physicist Jean-Baptiste Perrin veri¬ed Einstein™s result experimen-

tally. He measured x2 as a function of time t. Knowing the temperature T and viscosity ·

of the water, and radius a of the particle, he could obtain the value of the constant k. Using

this, he obtained a reasonably good value for Avogadro™s number (no. of molecules in a mole

of a substance).

• Einstein™s explanation of the Brownian Motion and its subsequent experimental veri¬-

cation by Perrin5 were historically important because they provided a convincing evidence

for the molecular theory of matter. In other words, they showed that atoms and molecules

are real physical objects. Skeptics who doubted their existence were silenced.

• Fluctuating force on a Brownian particle is but one example of a ¬‚uctuating physical

quantity even when the system is in equilibrium. Another example is a ¬‚uctuating current

in some electric circuits. Einstein™s work on the Brownian Motion laid the foundations of

the study of ¬‚uctuation phenomena as a branch of statistical mechanics.

We have reached the end of our story of the Brownian Motion. You must have realized

how a lowly pollen grain can tell us so much about the constitution of matter. Note that

nothing of this would have been possible without the inquisitive mind of the scientist. The

following quotation comes to my mind:

There is something fascinating about science. One gets such wholesale returns of conjec-

ture out of such a tri¬‚ing investment of fact ” Mark Twain (1835 - 1910).

9

Footnotes:

1 Sometimes it is wrongly stated that Brown observed irregular motion of the pollen grains

themselves. Secondly, he was not the ¬rst to notice this phenomenon. But he was the ¬rst to stress

its ubiquitousness and to rule out its explanations based on the so-called life force. As a result of

his work, this subject was removed from the realm of biology into the realm of physics.

2 However good a theory may appear, if its predictions do not agree with experimental data, it

is discarded.

3

Here we assumed that successive time intervals are very small compared with the

observation time but still large enough that the motion of the suspended particle in any

time interval can be considered to be completely independent of its motion in the preceding

time interval.

4

Einstein showed that it is practically impossible to measure vx and suggested that

2

experimentalists should rather measure x2 .

5

Perrin was honoured with the Nobel Prize for Physics in 1926, for this work.

10