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Random walk from Einstein to the present

Random walk from Einstein to the present. Thomas Spencer School of Mathematics. Robert Brown (1773-1858). Leading Scottish Botanist Explored the coast of Australia and Tasmania Identified the nucleus of the cell.

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Random walk from Einstein to the present

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  1. Random walkfrom Einstein to the present • Thomas Spencer • School of Mathematics

  2. Robert Brown (1773-1858) • Leading Scottish Botanist • Explored the coast of • Australia and Tasmania • Identified the nucleus of • the cell

  3. Around 1827 Brown made a systematic study of the “swarming motion” of microscopic particles of pollen. This motion is now referred to as Brownian movement. (Brownian motion). At first, “…I was disposed to believe that the minute spherical particles were in reality elementary units of organic bodies.”

  4. Brown then tested plants that had been dead for over a century. He remarks on the “vitality retained by these molecules so long after the death of the plant” Later he tested: “rocks of all ages … including a fragment of the Sphinx” Conclusion: origin of this motion was physical, not biological.

  5. Real Brownian Movement

  6. His careful experiments showed that motion was not caused by water currents, light, evaporation or vibration. He could not explain the origin of this motion. Many later experiments by others - Inconclusive. But by the late 1800’s the idea B-movement was caused by collisons with Invisible molecules gained some acceptance.

  7. Schematic Brownian movement

  8. Brownian Motion explained in 1905 with the work of Albert Einstein

  9. Title : “On the movement of small particles suspended in a stationary fluid as demanded by the laws of kinetic theory” Motivation: To justify the kinetic theory of atoms and molecules – and make quantitative predictions “In this paper it will be shown that according to the laws of molecular-kinetic theory of heat, bodies of a microscopically visible size suspended in a liquid must as a result of thermal molecular motions, perform motion visible under a microscope.”

  10. Although the idea of atoms goes back to the Greeks and the kinetic theory to Boltzmann and Maxwell, there were many skeptics and questions: Were atoms real? How many molecules in 18 grams of water ? – Avogadro number

  11. Einstein’s equations: B(t) = position of Brownian particle at time t. Distance [B(0), B(t)] = T=temperature, r =radius of particle , = viscosity k= Boltzmann constant

  12. Jean Perrin experimentally verified Einstein’s predictions In his letter to Einstein: “I did not believe it was possible to study Brownian motion with such precision” Accurate calculation of Avogadro number

  13. Mathematical interlude: What is a random walk?

  14. Random Walk Steps (moves) in all directions are equallylikely (No drift) Each step independent of previous step. How far does an N step Random Walk go? Distance from its starting point = N ~ t = time

  15. One Dimensional Random Walk

  16. 2D Random Walk

  17. 3D random walk

  18. Basic properties of Random Walk In 2D, Random walk is Recurrent: It returns to its starting point infinitely often. In 3D, Random walk is Transient: After some time, walk will Not return to its starting point.

  19. Fractal dimension = 2 This means that : In large cube of side L with L3 points inside, a random walk visits L2 points.

  20. Brownian motion = Limit of Random walk with infinitesimal,independent steps. Defined by Norbert Wiener (1920’s) A Brownian path, B(t), is a continuous function of time t, but it is very irregular. Crosses itself infinitely often in 2 or 3D In 4D, two Brownian paths do not cross. . In mathematics:

  21. Louis Bachelier(1870-1946) 1900 Thesis: “Theorie de la Speculation” .

  22. Major new Ideas and results: Market fluctuations described in terms of Brownian motion Brownian Motion has Normal distribution Martingale theory, Chapman-Kolmogorov eqn’s Bachelier-Wiener Process.

  23. Self-avoiding Walks or Polymers In 1940’s, Paul Flory, chemist, studied long chains of monomers – polymers Each monomer ~ step of the walk. Except: monomers cannot occupy the same space – excluded volume effect

  24. Polymer made of 500 monomers

  25. What is the diameter of polymer made of N monomers? Each polymer with N monomers equally likely In 2D: Diameter = C N3/4 ? Fractal dimension = 4/3 In 3D: Diameter = C N ,   .6 ?? Above 4D , polymer ~ random walk ,  = .5

  26. Self-Avoiding path with 20,000 steps

  27. Branched Polymer

  28. Branched polymer, N = 10,000

  29. Each branched polymer formed with N edges or monomers is assumed to be equally likely. Theorem (D. Brydges and J. Imbrie, 2002): In3D Diameter of BP = C N = # monomers. Supersymmetry used to prove dimensional reduction. Problem is unsolved in 2, 5, 6, 7 dimensions.

  30. SLE revolution in 2 Dimensions: Charles Löwner, 1920’s , studied 2D conformal mappings using differential equations. SLE = Brownian motion + Löwner’s equation Oded Schramm, Greg Lawler and Wendelin Werner (2000 – present) Solved: manyproblems in the geometry of Brownian paths, percolation, loop erased, Walks, CFT….

  31. The boundary of Brownian path in 2D has fractal dimension = 4/3. (LSW 2000)

  32. Driving in Manhattan or Quantum diffusion (Model due to John Cardy and others)

  33. Obstructions at street corners appear randomly with probability = p , 0<p<1 Driver is a robot and follows the streets and turns only at obstructions. This model is equivalent to a model of a quantum electron in 2D, interacting with Impurities (obstructions). Description of the model:

  34. p=0.5

  35. p=0.25

  36. Conjectures (for p >0): All paths eventually form a loop. (known for p>1/2) Electron is trapped – no conduction. If obstructions are rare, loops are extremely long. Paths behave like random walks for a very long time: Thus electron diffuses for a long time before it is trapped.

  37. Comments If p=1/10 the average length of the loop ~ 1040 Numerical computations are not reliable for p<1/4. Most paths are too long for modern computers to check whether a path eventually loops back. In 3D,expect that most paths do not close – motion is diffusive - like random walk.

  38. Breathing and Brownian Motion

  39. Lung surface has a complicated fractal structure. Think of oxygen molecules moving about through collisions like a Brownian path. What is the optimal shape of a surface for it to absorb oxygen most efficiently? Where would a Brownian molecule most likely strike a surface? (Harmonic measure)

  40. If you make the surface too rough, (fractal dimension too high), the Brownian paths will be unable to hit most of the surface. Jean Bourgain - Fractal dimension of a surface that BM can hit cannot be too close to 3. Tom Wolff - showed that there exist surfaces of fractal dimension bigger than 2 which are accessible to a Brownian path.

  41. Conjecture: The largest fractal dimension of a surface accessible to Brownian path is 2.5 (Peter Jones).

  42. Acknowledgements Thanks to: Joel Lebowitz and Michael Loss And to: Thomas Uphill and Michelle Huguenin

  43. (t) b a c via Conformal map B(at) a b c LÖWNER

  44. Some conventional wisdom: certain quantum field theories are equivalent to a gas of Brownian paths in 4D. Interaction occurs when the paths intersect. No interaction in 4D ? ? . Model is not interacting unless embedded in non-abelian gauge theory

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