How "the drunkard's walk" changed physics

Illustration for article titled How "the drunkard's walk" changed physics

Brownian motion was one of the many things explained by Einstein. It's the seemingly random 'jiggling' of particles caused by atoms hitting them. Under some circumstances, though, it's not so random. Find out about the strange patterns of Brownian motion, and how physicists have observed them for the first time.


Back in 1827, Robert Brown was observing pollen grains in water under a microscope. He was having a difficult time, since the little suckers kept jumping. Each grain moved in a random direction at a random time, not flowing all in one direction the way they would if a larger outside force was acting on them. Brown wrote about the tiny, random movements, and lucked out. Despite the fact that the phenomenon had been observed in one medium or another since ancient Greece, he ended up being the namesake of the motion. Although people came up with the correct explanation for it — the grains were taking constant hits from atoms and molecules too small to be seen — Einstein finally came up with a way to establish a way to measure how the grain moved.

Although he didn't follow the path of the individual movements of the grain, he showed that the grain took 'The Drunkard's Walk.' The Drunkard's Walk compares the random movement of molecules to the random movement of a drunk. This particular drunk is asked to walk ten steps from his original position. But he's so blotto that every step is in a random direction. The first step might be to one side, the next back to where he started, the next to the other side, and so forth. To take ten steps in a line is a task that spans length, and length is one dimension, so we only deal with one value. When the steps can be taken in any direction, the task turns from one that spans length to one that spans area. Area is two dimensions. To find an area, we multiply length times width. So if we want the drunkard to go ten steps from his starting point, we calculate that he will take ten times ten, or a hundred steps to do it. This is a sign of true random motion in two dimensions - the distance gone is the square root of the number of steps it takes to get there.

The motions of the molecules were too quick to count steps, but Einstein showed that, over any amount of time they moved from their starting points in the manner of The Drunkard's Walk. Give a single pollen granule one minute, and it will move a single unit of distance from its starting point. Give it four minutes and it will move two units from its starting point. Give it nine minutes and it will move three units. This same works if the unit of time is hours or days - though the pollen would require a large slide and would be a pain to watch. The distance it covers will always be proportionate to the square root of the time it's given. Brownian motion is a true Drunkard's Walk. It's motion in a random direction.

Oh, but it isn't. Einstein's paper isn't disputed, but it has been built on. Scientists observed that, under certain conditions, Brownian motion isn't completely random. When the densities of the fluid and the particle are similar, the motion of the particle affects the motion of the molecules in the fluid, which then affect the motion of the molecule. A dense enough molecule drags the fluid along with it. This subjects it to what is called 'hydrodynamic memory.' The flow of the fluid will encourage the previous motion, and discourage any different motion that the particle makes. When it is given a second kick along the direction of motion, it moves farther in one direction. When it is given a kick against the direction of motion, that kick is stifled by the fluid around it.

The most famous literary Brownian motion, of course, is in The Hitchhiker's Guide to the Galaxy. In it, the Heart of Gold spacecraft is powered by an 'infinite improbability drive,' which is an improbability generator suspended in 'Brownian motion' device - the example given is a cup of hot tea. The idea is the random motions of the particles in the tea work with the generator to let a spaceship travel everywhere in the universe simultaneously and get to the farthest reaches of the universe instantly. With this new kind of Brownian motion observed, there may be a new kind of engine called the 'destiny drive.' By suspending an improbability generator in a cup of liquid of the same density as the generator, people could get to where they're going instantaneously - but only if they were already moving in that direction.

Image: Francesco Bafs.

Via BUN and Physics World.



Stephan Zielinski

Er... not quite. The MEAN distance traveled from the origin will be proportional to the square root of time T. But individual particles will do as they damn well please— and in fact, the most likely distance traveled from the origin will be zero. (Statistics folks distinguish between a distribution's "mean"— what we call an "average" in everyday parlance— and its "mode", which is "The most likely result." For many important applications, the mode is at the mean— but it's not always the case. Here, the mean is proportional to sqrt(T), but the mode is zero.) More intuitively, if "The distance it covers will always be proportionate to the square root of the time it's given" held true, opening a bottle of perfume in a room would yield a shock front of perfume molecules proceeding outwards in a shell at a distance proportional to sqrt(T). In reality, you end up with a cloud that's thin on the edges and densest in the middle. (Until the walls become an issue, of course.)

Nor does the square root come in because we're talking about two dimensional motion; the mean distance traveled remains proportional to sqrt(T) even if we're talking about three dimensions. The actual relevant equation is the "diffusion equation", which I don't even want to try to render in a C-code like form— particularly since I'd just be copying the relevant bit out of Wikipedia: []