Einstein on Brownian Motion

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Einstein on Brownian Motion

• R. S. Bhalerao
• Department of Theoretical Physics
• TIFR, Mumbai
• University of Pune
• 12 November 2005

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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.)

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1905 Einsteins miraculous year

• Einsteins papers on
• Special Theory of Relativity
• Photoelectric Effect
• Brownian Motion
• published in 1905, brought
• about a revolution in physics.

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World Year of Physics 2005
http//www.tifr.res.in/ipa/Prog.htm
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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
  partys anti-Jewish activities.

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• Pacifist. Gentle. Modest. Deeply religious.
• Loved music. An accomplished violinist.
• One of the greatest scientific thinkers of all
  time.

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What is Brownian Motion?
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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

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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.

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A Thought Experiment (contd.)A Funny
TorchIt turns on only for a moment,
every 15 seconds.
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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
• t15 sec, at B
• t30 sec, at C
• t45 sec, at D
• You have no idea what is going on.

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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!

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What do you think was going on?
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• A drunken man wandering around aimlessly.
• That was easy. One does not need an Einsteins
  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.

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Random Walk in One Dimension

• ltxgt 0
• ltx2gt N, the number of steps
• (if each step of unit length)
• ltx2gt N l2
• (if each step of length l)
• xrms vltx2gt ? 0
• What does all this mean?

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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?

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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.

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• History of the Brownian Motion

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• He is happiest who hath power to
  gather wisdom from a flower.
• Mary Howitt (1799 -1888)
• (English poetess)

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• Now I want to describe
• a real (not a gedanken) experiment.

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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.
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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?

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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.

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Result of an actual experiment

• Particle positions were recorded at intervals of
  30 sec.

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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.

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• 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

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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.

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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.

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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.

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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
• ltx2gt kTt / (3p?a)
• k Boltzmann constant, T temperature,
• t elapsed time, ? viscosity of the liquid,
  
• a radius of the suspended particle.

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Jean Baptiste Perrin
(1870-1942) French Physicist Nobel
Prize 1926In 1908, he verified Einsteins
result experimentally and obtained a reasonably
good value for Avogadros number (no. of
molecules in a mole of a substance).
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Einsteins approach to the Brownian Motion
problem

• He based his analysis on the osmotic pressure
  rather than on the equipartition theorem.
• He identified ltx2gt 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.

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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 !

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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.

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References

• Einsteins 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)

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• 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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• Thank you