Walking Wild II

In the 1920s, Norbert Wiener transformed a random walk (see "Walking Wild I"), which is made up of discrete steps, into a mathematical model suitable to represent Brownian motion (see "Quivering Particles I"). He did it by making the steps or time intervals between steps infinitesimally small.

In Wiener's rigorous approach, once the position of a particle is established at the start, its position at any later time is governed by a Gaussian curve, as it is in Albert Einstein's physical model of Brownian motion (see "Quivering Particles II").

Wiener also proved that although the path of a Brownian particle is continuous, at no point is it smooth. Such a curious, incredibly jagged mathematical curve actually makes physical sense because a particle in Brownian motion can't jump instantaneously from one position to another, so its path must be continuous.

At the same time, as Jean Baptiste Perrin noted, erratic changes in direction appear to take place constantly, so you might expect the path to consist entirely of sharp corners. In fact, a two-dimensional Brownian trajectory wiggles so much that it ends up filling the entire area over which the motion occurs.

Two-dimensional Brownian motion.

With Brownian movement put on a solid mathematical footing by Wiener and others who followed his lead, such abstract formulations began to play a significant role in the creation of models of random phenomena.

Such models were used to represent the diffusion of heat through a metal, the spread of flu epidemics via the random walks of microbes, the structure of background noise and static affecting radio signals, the transport of perfume molecules from source to nose, and even the spread of rumors.

Wiener himself applied his model of Brownian movement to the problem of electronic interference, which disturbs the transmission of radio signals and causes the type of static heard on AM radio between stations.

Because the static has a strongly random character, Wiener was able to use the mathematics of Brownian motion to design an electronic filter to separate the signal from the background noise. Applied to the development of radar during World War II, his results were long kept a military secret.

It is interesting to note that applications of Brownian motion often began with the study of biological processes. The law of diffusion and models of the spread of heat in a material, for instance, initially arose from studies of heat generation in muscles.

Next: Walking Wild III