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Random walk from Einstein to the present Thomas Spencer School of Mathematics

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

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

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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” 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” Later he tested: “rocks of all ages … including a fragment of the Sphinx” Conclusion: origin of this motion was physical, not biological. Conclusion: origin of this motion was physical, not biological.

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Real Brownian Movement

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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.

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Schematic Brownian movement

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Brownian Motion explained in 1905 with the work of Albert Einstein

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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.”

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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

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

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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

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What is a random walk? What is a random walk? Mathematical interlude:

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Steps (moves) in all directions are equally likely (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 N ~ t = time Random Walk

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One Dimensional Random Walk

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2D Random Walk

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3D random walk

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

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Fractal dimension = 2 This means that : This means that : In large cube of side L with L 3 points inside, a random walk visits L 2 points.

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In mathematics : 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..

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

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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.

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

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Polymer made of 500 monomers

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

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Self-Avoiding path with 20,000 steps Self-Avoiding path with 20,000 steps

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Branched Polymer

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Branched polymer, N = 10,000

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Each branched polymer formed with N edges or monomers is assumed to be equally likely. Theorem (D. Brydges and J. Imbrie, 2002): In 3D Diameter of BP = C N = # monomers. Supersymmetry used to prove dimensional reduction. Problem is unsolved in 2, 5, 6, 7 dimensions.

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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: many problems in the geometry of Brownian paths, percolation, loop erased, Walks, CFT….

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

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

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p=0.5

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p=0.25

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Conjectures (for p >0): All paths eventually form a loop. 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.

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Comments If p =1/10 the average length of the loop ~ 10 40 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.

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Breathing and Brownian Motion Breathing and Brownian Motion

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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)

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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.

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

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

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ac b (t) via Conformal map abc B(at) LÖWNER

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Some conventional wisdom: c ertain 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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