Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,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,p):,0,q,p),Not always the case,e.g.dust particle in wind,P(,q,r,),=,|q|/p,2,-,p q p,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?,