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### Einstein on Brownian Motion

R. S. Bhalerao

Department of Theoretical Physics

TIFR, Mumbai

University of Pune

12 November 2005

20th Century – 3 major revolutions in our physical picture of the world

- Theory of Relativity
- Quantum Theory
- Nonlinear Dynamics and Chaos

A single physicist - A. Einstein - in a

single year - 1905 - laid foundation stones of the first two of these revolutions.

Moreover, in this same year, he provided fundamental new insights into two other areas. (Full-time job at Patent Office.)

1905: Einstein’s miraculous year

Einstein’s papers on

- Special Theory of Relativity
- Photoelectric Effect
- Brownian Motion

published in 1905, brought

about a revolution in physics.

World Year of Physics – 2005http://www.tifr.res.in/~ipa/Prog.htm

Albert Einstein (1879-1955)

- b. Ulm, Germany. Jewish parents.
- Went to school in Munich. Excelled in math.
- 1896-1900: ETH, Zurich, Switzerland.

Graduated in Theoretical Physics & Math.

- 1905: Ph.D., University of Zurich.

Also published a set of very famous papers.

- 1914: Director of a new inst. in Berlin, Germany.
- 1916: Published the General Theory of Relativity.
- 1921: Nobel Prize.
- 1933: Migrated to USA because of the Nazi party’s anti-Jewish activities.

Pacifist. Gentle. Modest. Deeply religious.

Loved music. An accomplished violinist.

One of the greatest scientific thinkers of all time.

Let us do a “thought experiment”

What is a “thought experiment”?

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.

German word:Gedankenexperiment

A Thought Experiment:

Imagine a dark, cloudy, moonless night.

Suppose: power outage in the entire city.

You are sitting in your 4th floor apartment thinking and worrying about your physics test tomorrow.

Suddenly: a commotion downstairs.

You somehow manage to find your torch and rush to the window.

A Thought Experiment (contd.)A Funny Torch:It turns on only for a moment, every 15 seconds.

Here is what you see whenever the torch lights up:

- A man standing in the large open space in front of your building.
- t= 0 sec, at A
- t=15 sec, at B
- t=30 sec, at C
- t=45 sec, at D
- …

You have no idea what is going on.

When

- You mark his positions on a piece of paper.
- Connect point A to B, B to C, C to D, and so on, by straight lines.
- What do you see? A zigzag path!

A drunken man wandering around aimlessly.

That was easy. One does not need an Einstein’s IQ to figure that out.

Physicists: (an almost) random walk in 2D.

(2D because: length and breadth)

Imagine a random walk in 1D & then in 3D.

Random Walk in One Dimension

- <x> = 0
- <x2> = N, the number of steps

(if each step of unit length)

- <x2> = N ℓ2

(if each step of length ℓ)

- xrms = √<x2> ≠ 0
- What does all this mean?

Another Thought Experiment

Suppose you are sitting in a big stadium, watching a game of football, being played between two equally good teams.

Suppose the players are invisible: nothing except the ball is visible.

Open your eyes once every 15 sec.

What will you see?

Is the drunk?

- The ball moves almost like the drunkard.
- Is it drunk? Of course, not.
- It moves that way because it is being hit

repeatedly by the players in the two teams.

- This is another example of an (almost) random walk in 2D.

What you learnt above is the ABC of the

branch of physics called Statistical Mechanics.

“He is happiest who hath power to gather wisdom from a flower.”

… Mary Howitt (1799 -1888)

(English poetess)

Now I want to describe

a real (not a gedanken) experiment.

Robert Brown (1773-1858) (Scottish Botanist)In 1827, he observed through a microscopepollen grains of some flowering plants. To his surprise, he noticed that tiny particles suspended within the fluid of pollen grains were moving in a haphazard fashion.

If you were Robert Brown

- How would you understand this observation? (Remember, you are in 1827!)
- Would you suspect that the pollen is alive?
- Would you get excited at the thought that you may have discovered the very essence

of life or a latent life force in every pollen?

- What other experiments would you perform to test your suspicions?

This is what Mr. Brown did:

He repeated his experiment with other fine

particles including the dust of igneous rocks, which is as inorganic as could be.

He found that any fine particle suspended in water executes a similar random motion.

This phenomenon is now called Brownian Motion.

Result of an actual experiment

Particle positions were recorded at intervals of 30 sec.

A Quiz

- Scientists in the 19th century were puzzled by this mysterious phenomenon.
- 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 1st thought experiment.
- Why then is it moving? And why in an erratic way? Think …
- Want a hint? Recall our 2nd thought expt.

If you have not already guessed, here is the rational explanation for the mysterious jerky movement of tiny particles suspended in fluids, which made Mr. Brown famous:

Basic Understanding

- Sizes (radius or diameter)

Suspended particle: a few microns (10-6 m)

Atom: 10-10 m

Water molecule: somewhat larger

- Thus the suspended particle is a monster, about 10,000 times bigger compared to a water molecule.

Basic Understanding (contd.)

- Numbers: A spoonful of water contains about 1023 water molecules.
- Speeds: They are perpetually moving in different directions, some faster than others.
- As they move, they keep colliding with each other, which can possibly change their speeds and directions of motion.

Basic Understanding (contd.)

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 net force on it keeps fluctuating in time and it keeps getting kicks in the direction of the net instantaneous force. The end result is that its position keeps changing randomly.

Contribution

- Qualitative vs quantitative understanding:

He presented a detailed mathematical theory.

- Crucial insight: Brownian motion as the

microscopic process responsible for diffusion

on a macroscopic scale.

- Using ideas from Statistical Mechanics, he

showed that the mean-square displacement is

<x2> = kTt / (3πηa)

k = Boltzmann constant, T = temperature,

t = elapsed time, η = viscosity of the liquid,

a = radius of the suspended particle.

Jean Baptiste Perrin (1870-1942) French Physicist : Nobel Prize 1926In 1908, he verified Einstein’s result experimentally and obtained a reasonably good value for Avogadro’s number (no. of molecules in a mole of a substance).

Einstein’s approach to the Brownian Motion problem:

- He based his analysis on the osmotic pressure rather than on the equipartition theorem.
- He identified <x2> of suspended particles rather than their velocities as suitable observable quantities.
- He simultaneously applied the molecular theory of heat and the macroscopic theory of dissipation to the same phenomenon.

Importance of Contribution

- It provided a convincing evidence for the molecular theory of matter: It showed that atoms and molecules are real physical objects. Skeptics who doubted their existence were silenced.
- It laid the foundations of the study of fluctuation phenomena as a branch of Statistical Mechanics.

Historically important !

Moral of the Story

- Even a lowly pollen grain can tell us a lot about the constitution of matter.
- Nothing of this would have been possible without the inquisitive mind of the scientist.

References:

- Einstein’s Miraculous Year

edited by John Stachel, Princeton, 1998

- 100 Years of Brownian Motion

P. Hanggi and F. Marchesoni

cond-mat/0502053

- Brownian Motion: Theory & Experiment

K. Basu and K. Baishya

Resonance Vol. 8, No. 3, 71-80 (2003)

“There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”

… Mark Twain (1835-1910)

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