# DoITPoMS

Random walk

If we consider an atom undergoing diffusion, we find that we cannot precisely predict its motion. Where the atom is from time to time is essentially random. If there is a preferred direction of motion (perhaps caused by an electric field) then from time to time the atom is more likely to move in that direction. Similarly, if we consider many atoms evenly distributed undergoing diffusion, then their average motion is zero. If there is a preferred direction of motion, we find that there will be a drift in that direction. This random motion can be modelled using a random walk.

## One-dimensional random walk

Imagine an atom sitting in an atomic site. The atom will oscillate n times per second, corresponding to a vibration frequency ν. The jump distance, l, is the distance between atomic sites. When left for a time, t, the atom will make a succession of jumps, randomly left or right, and will end up a distance, d, from its starting point. This is known as a “random walk”

For a random walk, the mean distance moved, ,is proportional to the root-mean square of the number of jumps made, and therefore to the square root of time:

It is significant that we deal with the mean distance travelled. Since the movement of the atom is governed by probabilities, there is a statistical distribution of distances travelled by atoms. For a single atom we cannot know where it will be - only assign probabilities to its potential locations.

This simulation shows the random walk of an atom along a one-dimensional lattice. When there is no bias, there is an equal probability that the atom will move in either direction. If a bias is applied the atom has a greater probability of moving to the right, and statistically is likely to drift in this direction with time.