Brownian motion is one of Einstein’s less popularly known proofs.

When you look at a particle under the microscope, it moves. But it does not move in one direction or according to any seen force. It seems to move in completely random directions. It’s a form of stochastic drift.

A particle in water is struck by water molecules. The molecules are constantly moving and bounce off the particle. This applies kenetic energy to push the particle in that direction. There are a lot of water molecules are striking the particle from a number of directions, which accounts for the seeming randomness of the drifting particle.

Here’s an applet demonstrating the principle.

While it is possible to measure the kenetic energy of any individual collision, it is futile to measure the astronomical number of collisions per minute to account for Brownian motion is a realistic timeframe.

The way around this is a mathematical probability model. This uses a partial differential equation to simulate the random walk of the particle. This describes the probability of movement at any point.

I was thinking about this today. To really understand Brownian Motion, you have to go back to your physics textbooks from highschool and do the math by hand. Simply being told what the process is does not mean we understood how it works.

Here’s one example of the mathematics.

It’s not a particularly hard formula at all. But it’s still difficult to describe in words.

This gives us a way of measuring probabilities of Stochastic Drifts. Normally, we would be fooled by randomness into seeing “patterns” which have no underlying cause.