## Brownian Motion Simulations

Brownian Motion Applet

Copyright © 1998 by Paul O. Lewis

[Source code]

This applet shows a panel of 16 random walk simulations. In each simulation, two particles begin random walks at the same point. At regular intervals, the applet polls each simulation to determine how far apart the two particles are at that point in time. It then computes the sample variance of these 16 differences to illustrate that the variance of the distance between two particles undergoing random walks increases linearly with time. The variance is actually expected to equal the amount of time (measured in forward steps) separating the two particles. Thus, if both particles have moved two steps forward, the variance of the distance separating them should equal 4.

Early on in a simulation the distance between the particles is confined to a fairly narrow range of possibilities, whereas late in the game the distance between them becomes much more difficult to predict. Although these random walks represent a simplistic special case, they illustrate nicely the major features of the Brownian motion model used by Joe Felsenstein in his paper (Felsenstein, J. 1985. Amer. Nat. 125:1-15) on independent phylogenetic contrasts. His model assumes that continuous characters begin at the same value at the point at which two lineages become separated (i.e., a speciation event). The model also assumes that the difference in trait values of the two lineages becomes more unpredictable as time goes on (that is, the variance of the difference increases linearly with time).

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