Dealing with random things

In physics, just like elsewhere in life, there can be more than one way to tackle a problem. An example I’ve been thinking about recently is in the computer modelling of random processes (which is essentially what my research is about). Rather than talk about neurons and what causes them to fire or not-to-fire, I’ll draw on a more well-known example, the drunkard’s walk.

I’m not sure who coined the phrase, and whether the term ‘drunkard’ refers to any particular physicist, but in simple terms the process is this: 

A physicist emerges from a bar late at night and has absolutely no idea which way home is. Every step he takes is in a completely random direction, and each step is uncorrelated with the previous one. (In other words, he forgets which way he was travelling in after every step.)  Where does he end up after ten minutes (assuming he is still on his feet)?

Of course, we don’t know the answer to this question – his movements are random – but we can say some things about his movement. First of all, on average, he will go no-where. He is just as likely to take a step northwards as southwards, so the mean position will be at the door of the bar. We could run a computer model here. Start lots of physicists at the door of the bar, and, at each time step, move each one, independently, either a step northwards or a step southwards. After several time steps, look where they are, on average. The centre of their distribution will be at the starting point.  BUT, the average distance of a physicist from the door will increase as time increases (that is, the distribution of the physicists spreads out). It can be shown (as Einstein did for Brownian Motion)  that the distance increases in proportion to the square root of time. If you quadruple the number of steps, the average distance from the door will double.

That’s a simple example, and realistic ones are more compllicated, usually with the motion being partly deterministic, or each step not being uncorrelated with the previous. But similar ideas apply. Buried in the above paragraph, I’ve hinted at two different approaches for considering this problem. One is simulating lots of examples – i.e. write down an equation of motion for a physicist, and simulate it on the computer lots of times.  For such a walk, this would be an example of a Langevin equation, after Paul Langevin, who studied Brownian Motion. (Technically, that’s not quite true, because the random part isn’t gaussian, but we could formulate the problem so it is, but that’s harder to describe.)

The other is considering the distribution of probability. Instead of writing down an equation for each physicist, we construct an equation for the probability of finding a physicist at a given point in space, and describe how that probability changes with time. This is called the Fokker-Planck equation. The two equations are equivalent descriptions of the same thing – from the Langevin equation you can i derive the Fokker-Planck equation. Which you choose to work with depends a bit on what you are trying to do.

So we have the same process, but we can use either method to describe it. That’s quite often the case in physics. Another nice example is the equivalence between Schroedinger’s equation and the Feynman path integral in quantum mechanics, but I’ll leave that to another blog entry.


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