I found the discussion on random walks quite interesting. It is such a simple concept; I can’t believe it took Einstein to realise it is a satisfactory explanation for Brownian motion. I think this demonstrates the lack of statistical intuition that most people, even scientists, suffer from.
What I think people fail to appreciate is that in a process that is that consists of steps that are independent, the steps are genuinely independent of each other. Each step has absolutely no idea what the outcomes of the events preceding it were.
Say a person flips a fair coin five times, and gets tails every time. If we were asked to predict what the result of the next flip is, at least part of us would be tempted to say a heads is more likely. An average person might justify this thought as the coin “being due for a heads”. However, we know that the result of the next flip is completely independent of the previous flips. The coin does not know that it’s “due for a heads”. It has the same chance as it always did of resulting in a heads when flipped.
Assuming that the next trial in a series of independent events will be more likely to result in a value that brings the average of the results towards the mean expected outcome of all the events previous, rather than the known likelihood of the results of the event, is called the Monte Carlo Fallacy. When this fallacy is avoided, it is much easier to accept that random motions can result in a visible net displacement, if the observer waits long enough.
What I found even more interesting is that the random walk has structure on all length scales. The idea of an object with structure on all length scales is impressive. It is like a fractal. Like a fractal, whose total perimeter can be calculated, the length of a random walk can also be calculated. It seems necessary that the path of a molecule should have structure on all length scales though. Considering a single molecule in a very large sample of water, like a lake, or an ocean, or the air molecules in an atmosphere, then if enough time passed, the particle should have the possibility of being anywhere in the mixture. The position of the particle should not be restricted to a subsection of the space, because there is nothing to enforce that restriction. So the path traced by the particle has structure even on the scale of the size of the medium it is in. There should be no restriction on the length scale of the path of a particle, other than the size of the medium that the particle is in.
I suspect that the concept of the random walk has not been fully explored, and I think that this idea will be used again in future chapters.
This has always bothered me about statistics. I guess my brain has the Monte Carlo fallacy ingrained. How do you reconcile the independence of single measurements with the knowledge that the distribution of results should be 50% heads and 50% tails?
ReplyDeleteI wanted to actually raise up a point concerning that idea that each step is independent of the previous steps. We know that brownian motion is affected by the collisions that one particle experiences from the surrounding particles, so would not the previous collision of all the particles have influence upon the outcome of the next movement? Because I mean that obviously the next motion of particle can differ if it were to have been going left or right previous to the collision to change its motion.
ReplyDeleteAlso, just on a side note about how it took Einstein to explain brownian motion, I do find it quite funny that he just dropped his thesis in order to explain it, or as it were said in the textbook, one of his "distractions".
At the end of the chapter it talks about how the independence assumption isn't always valid. Saying how a bullet fired into water doesn't lose the inital motion after the first molecular collision. The bullet however is far from equilibrium in terms of the water which results in the deviation. So it could be said that within an equilibriated system the assumption that collisions are independent will hold.
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