Brownian%20Motion%20and%20Diffusion%20Equations

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Brownian Motion and Diffusion Equations
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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

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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

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Histogram of 1000 p.5 random walks with 15 steps
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Random Walks in Several Dimensions

• Consider a particle that moves r distance in a d
  dimensional space with every step.
• A PDF p(r) determines the motion
• We assume p(r) is uniform (ie p(r) 1/(2p) 0 lt
  q lt p)
• Not always the case, e.g. dust particle in wind

P(q,r ) q / p2, -p lt q lt p
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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

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• Questions???