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  In 1905, Einstein developed a mathematical model for Brownian Motion

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…

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

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/(2  Not always the case, e.g. dust particle in wind P(r) =  ^2, - 

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

 Questions???