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Brownian Motion for the School-Going Child

R.S. Bhalerao
Tata Institute of Fundamental Research, Mumbai, India


I. Introduction
arXiv:physics/0412132 v1 21 Dec 2004




Let us do a “thought experiment”. What is a thought experiment? It is an experiment
carried out in thought only. It may or may not be feasible in practice, but by imagining
it one hopes to learn something useful. There is a German word for it which is commonly
used: Gedankenexperiment. It was perhaps A. Einstein who popularized this word by his
many gedankenexperiments in the theory of relativity and in quantum mechanics.
Coming back to our thought experiment: Imagine a dark, cloudy, moonless night, and
suppose there is a power failure in the entire city. You are sitting in your fourth ¬‚oor
apartment thinking and worrying about your physics test tomorrow. Suddenly you hear a
commotion downstairs. You somehow manage to ¬nd your torch and rush to the window.
Now suppose your torch acts funny: it turns on only for a moment, every 15 seconds. Initially,
i.e., at time t = 0 seconds, you see a man standing in the large open space in front of your
building. Before you make out what is happening, your torch is o¬. Next time it lights up,
i.e., at t = 15 sec, you see him at a slightly di¬erent location. At t = 30 sec, he is somewhere
else and has changed his direction too. At t = 45 sec, he has again changed his location and
direction. You have no idea what is going on. But, you continue observing him for some
more time. When the lights come back, you mark his positions on a piece of paper (see Fig.
1). At t = 0, he is at point A, at t = 15, he is at B, at t = 30, he is at C, and so on. Connect
point A to B, B to C, C to D, and so on, by straight lines. (Go ahead, grab a pencil and do
it.) What do you see? A zigzag path.




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What do you think was going on? Did you say “a drunken man wandering around
the square”? Right. That was easy. One does not need an Einstein™s IQ to ¬gure that
out. In physicists™ language, the above is an example of a random walk in two dimensions:
two dimensions because the open area in front of your building has a length and a breadth.
(Strictly speaking, a walk can be said to be random if the direction of each step is completely
independent of the preceding step. For simplicity, the steps may be taken to be of equal
length.) Before you read further, close your eyes and imagine a random walk in 1 dimension
and then a random walk in 3 dimensions.




Figure 1




E


H
G
B
F




D I

C

A




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Random walk in one dimension:
Here is an experiment you can do yourselves. You will need a plain paper, a ruler, a
pencil, a one-rupee coin, a small stone and a lot of patience. Draw a number line with
markings at ’10, ’9, ..., 0, ..., 9, 10. Place the stone at 0. Toss the coin. The rule is, if it
is heads (H), the stone is moved one place to the right and if it is tails (T), it is moved one
place to the left. For example, if you get H, H, T, ..., the stone moves from 0 to 1, 1 to 2, 2
to 1, ... . Toss the coin 10 times. Note the ¬nal position of the stone. Call it x1 . Obviously,
’10 ¤ x1 ¤ 10.
Replace the stone at 0 and repeat the experiment. Again note the ¬nal position of the
stone. Call it x2 . Obviously, x2 may or may not be equal to x1 .
If you were to repeat this experiment a very large number of times, say 1000 times, and
then take the average (¯) of x1 , x2 , x3 , ..., x1000 , what result do you think you will get?
x
Since each toss of the coin is equally likely to result in a H or a T, x1 , x2 , x3 , ..., x1000 will
be distributed symmetrically around the origin 0. Hence x is most likely to be zero.
¯
Interestingly, however, the average (x2 ) of x2 , x2 , x2 , ..., x2 , will not be zero, since
1 2 3 1000
these are all non-negative numbers. In fact, x2 turns out to be equal to the number of times

you toss the coin in each experiment, which is also equal to the number of steps (N) in the
1/2
= N 1/2 . Since the
random walk. (This is 10 in our experiment.) Thus x2 = N or x2
left-hand-side is the square root of the mean (= average) of the squares, it is called the rms
displacement and is denoted by xrms . Thus xrms = N 1/2 .
What is the meaning of the statement x = 0, but xrms = N 1/2 ? It means, in a random
¯
walk, the object is as likely to be found on one side of the starting point as on the other,
making x vanish. But at the same time, as the number of steps increases, the object is likely
¯
to be found farther and farther from the starting point.
Equivalently, imagine 1000 drunkards standing at the origins of 1000 parallel lines, and
then starting simultaneously their random walks along these lines. If you observe them after
a while, there will be nearly as many of them to the right of the centres as there are to the
left. Moreover, the longer you observe them, the farther they are likely to drift from the
centre.
Conclusions: (a) x = 0. (b) xrms = N 1/2 if each step is of unit length. (c) xrms = N 1/2 l
¯
if each step is of length l.


Let us perform another thought experiment. Suppose you are sitting in a big stadium,
watching a game of football or hockey, being played between two equally good teams. As
in the previous thought experiment, you mark on a piece of paper the position of the ball
every 15 seconds, and then connect these positions in sequence. What do you see? Again
a zigzag path. The ball is moving almost like the drunken man. Would you say the ball is
drunk? Of course, not. The ball is moving that way because it is being hit repeatedly by


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the players in two competing teams. This is another example of an (almost) random motion
in two dimensions.
Want to impress someone? Remember this: Random processes are also called stochastic
processes. Chance or probability plays an essential role in these processes.
What you learnt above is the ABC of the branch of physics, called Statistical Mechanics.

II. History

He is happiest who hath power to gather wisdom from a ¬‚ower ” Mary Howitt (1799 - 1888).


Now I want to describe a real (not a gedanken) experiment. Robert Brown was a British
botanist. In 1827, he observed through a microscope pollen grains of some ¬‚owering plants.
To his surprise, he noticed that tiny particles suspended within the ¬‚uid of pollen grains
were moving in a haphazard fashion.1 If you were Robert Brown, how would you understand
this observation? (Remember, science in 1827 was not as advanced as it is today. Many
things written in your science textbook were not known then.) Would you suspect that the
pollen grain is alive? Or would you get excited at the thought that you have discovered
the very essence of life or a latent life force within every pollen? Or perhaps this is just
another property of organic matter? What other experiments would you perform to test
your suspicions?
Brown repeated his experiment with other ¬ne particles including the dust of igneous
rocks, which is as inorganic as could be. He found that any ¬ne particle suspended in water
executes a similar random motion. This phenomenon is now called Brownian Motion. Figure
2 shows the result of an actual experiment: the positions of the particle were recorded at
intervals of 30 seconds. (From J. Perrin, Atoms, D. Van Nostrand Co., Inc., 1923.) Similar
observations were made for tiny particles suspended in gases.
Scientists in the 19th century were puzzled by this mysterious phenomenon. They at-
tempted to understand it with the help of ideas such as convection currents, evaporation,
interaction with incident light, electrical forces, etc. But they had no satisfactory explanation
for it. With your knowledge of modern science, can you provide a rudimentary explanation?
Obviously, the suspended particle is not moving on its own unlike the drunkard in our ¬rst
gedankenexperiment. Why then is it moving? And why in an erratic way (see Fig. 2)?
Think, before you read further.




Want a hint? Recall our second gedankenexperiment.




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Figure 2
15 microns




III. Basic Understanding

If you have not already guessed, here is the rational explanation for the mysterious jerky
movement of tiny particles suspended in ¬‚uids, which made Mr. Brown famous:
• The size ” the radius or diameter ” of the suspended particle is roughly of the order
of a few microns (1 micron = 10’6 m). The size of an atom is of the order of 10’10 m.
The size of a water molecule (H2 O) is somewhat larger. Thus the suspended particle is a
monster, about 10000 times bigger compared to a water molecule. Also note that a spoonful
of water contains about 1023 water molecules. (The atomic or molecular theory of matter
which says that matter consists of atoms and molecules, is well-established today. It was
not so in 1827!)
• You also know that molecules of water (or molecules in any sample of a liquid or gas)
are not at rest. They are perpetually moving in di¬erent directions, some faster than others.
As they move, they keep colliding with each other, which can possibly change their speeds
and directions of motion.
• Now you can very well imagine the fate of the particle unfortunate enough to be placed
in the mad crowd of water molecules. The poor fellow is getting hit, at any instant, from
all sides, by millions of water molecules. The precise number of water molecules which hit
the particle at any instant, the exact points where they hit it, their speeds and directions
” all keep changing from time to time. (It is practically impossible and also unnecessary
to have this information.) This results in a net force which keeps ¬‚uctuating in time, i.e.,
its magnitude and direction keep changing from one instant to another. The particle keeps
getting kicks in the direction of the instantaneous net force. The end result is that its position

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keeps changing randomly as in Fig. 2.
IV. A Quiz

Ready for a quiz? Here are a few easy questions:
(1) Imagine the game of football played by invisible players. (Nothing except the ball is
visible.)
(2) See Fig. 2. If the positions of the particle were recorded every 60 seconds, instead of
every 30 seconds, how will the pattern look like?
(3) How will the Brownian Motion be a¬ected if (a) water is cooled, or (b) instead of
water a more viscous liquid is taken, or (c) the experiment is done with a bigger particle?

V. Einstein™s Contribution

You now have a qualitative understanding of the Brownian Motion. But that is usually
not enough. Scientists like to develop a quantitative understanding of a phenomenon. This
allows them to make precise numerical predictions which can be tested in the laboratory.2
For example, one would like to know how far (on an average) will the particle drift from its
initial position in say 10 minutes? How will its motion be a¬ected if the water is cooled by
say 5 C, or if the viscosity of the liquid is increased by 10%, or if the particle size is exactly
doubled?
In 1905, Einstein published a detailed mathematical theory of the Brownian Motion,
which allowed one to answer these and many other interesting questions. How did he do it?
I will only give you a ¬‚avour of what is involved.




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What is an ensemble?:
Recall that chance or probability plays an important role in random processes. Hence,
in the Introduction, when we discussed the random walk (see the box), you were asked to
do the experiment 1000 times and then average the results. If you do it only a few times,
x may not vanish and x2 may not equal 10. If you are too lazy to do the experiment 1000
¯
times, there is a way out: Get hold of 1000 friends of yours, ask each of them to prepare a
similar experimental set-up, and let each of them do the experiment only once. If you then
take the average of the results obtained by them, you will ¬nd x ≈ 0 and x2 ≈ 10.
¯
Similarly, if you observe the Brownian Motion of a particle only a few times, based
on these observations, you would not be able to make quantitative statements about its
average behaviour. You need to repeat the experiment a large number of times and take
the average of all the results. Alternately, you could prepare a large assembly of identical
particles, observe each of them once under identical experimental conditions, and then take
the average.
Physicists use the word ensemble to describe such an imaginary assembly of a very large
number of similarly prepared physical systems. The average taken over all the members of
the ensemble is called an ensemble average. In the following, when we talk about an average
behaviour of a Brownian particle, we mean an ensemble average.




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Let us ask ourselves a few simple questions about the average behaviour of a tiny particle
suspended in a liquid. Taking the initial location of the particle as the origin, imagine
drawing x, y and z axes in the liquid. Let (x, y, z) denote the coordinates of the particle.
x : Where will the particle be after some time? In other words, what will be the values
¯
of x, y and z after some time? (Remember the overhead lines denote ensemble averages.)
¯¯ ¯
Since this is a case of a random walk3 in 3 dimensions, x = y = z = 0.
¯¯¯
vx : What will be the average velocity of the particle parallel to the x axis? When we
¯
talk of the velocity of a particle, we have two things in our mind: its speed (fast or slow) and
its direction of motion. Since the particle is as likely to move in the positive-x direction as in
the negative-x direction, vx is as likely to be positive as negative. Hence vx = 0. Similarly,
¯

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