Brownian Motion

The function brownian.motion() has illustrated the phenomenon of Random Walk on the 2D plane. For a point (x, y), its next location is (x + rnorm(1), y + rnorm(1)).

# create an HTML animation page of Brownian Motion
# store the old option to restore it later
oopt = ani.options(interval = 0.05, nmax = 100, ani.dev = png, ani.type = "png",
    title = "Demonstration of Brownian Motion",
    description = "Random walk on the 2D plane: for each point (x, y), 
    x = x + rnorm(1) and y = y + rnorm(1).")
ani.start()
opar = par(mar = c(3, 3, 1, 0.5), mgp = c(2, .5, 0), tcl = -0.3,
    cex.axis = 0.8, cex.lab = 0.8, cex.main = 1)
brownian.motion(pch = 21, cex = 5, col = "red", bg = "yellow")
par(opar)
ani.stop()
ani.options(oopt)

The animation is like this:

• I find this Java applet is much more interesting than my function: Einstein's Explanation of Brownian Motion

If you use the lattice package, don't forget to print() your plots, otherwise graphics devices like png() will not be able to record your images. Here is an example:

library(animation)
library(lattice)
bm.mat = data.frame(x = rnorm(10), y = rnorm(10))
oopt = ani.options(interval = 0.05, nmax = 100, 
    title = "Demonstration of Brownian Motion",
    description = "Random walk on the 2D plane: for each point (x, y), 
    x = x + rnorm(1) and y = y + rnorm(1).")
## Brownian Motion in lattice
for (i in 1:ani.options("nmax")) {
    print(xyplot(y ~ x, bm.mat, xlim = c(-10, 10), ylim = c(-10, 10), pch = 19))
    bm.mat = bm.mat + data.frame(x = rnorm(10), y = rnorm(10))
}
ani.stop() 
ani.options(oopt)