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Brownian Motion and Diffusion Equations

Brownian Motion and Diffusion Equations. History of Brownian Motion. Discovered by Robert Brown, 1827 Found that small particles suspended in liquid moved about randomly Guoy discovered that particle motion was caused collisions of molecules

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Brownian Motion and Diffusion Equations

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  1. Brownian Motion and Diffusion Equations

  2. History of Brownian Motion • Discovered by Robert Brown, 1827 • Found that small particles suspended in liquid moved about randomly • Guoy discovered that particle motion was caused collisions of molecules • In 1905, Einstein developed a mathematical model for Brownian Motion

  3. Discrete Model Brownian Motion • Consider N discrete, independent steps in which a particle will move right with probability p, or to the left with probability 1-p. • Clearly, the number of right steps the particle takes is binomially distributed with parameters p and N. The number of left steps is binomially distributed with parameters 1-p and N. • The final position of the particle, the number of right steps minus left steps, has an expectation N(1-2p) • Binomial curve approximates to normal curve for large values of N and many trials, yielding…

  4. Histogram of 1000 p=.5 random walks with 15 steps

  5. Abstract Construction of a Brownian Motion • A function X(t) is a Brownian Motion iff: • 1) The mechanism producing random variations does not change with time. (ie, identical motions) • 2) All time intervals are mutually independent • 3) X(0) =0 and X(t) is a continuous function of t

  6. Questions???

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