Tuesday, February 11, 2014

The geometry of drifting apart

Why do point particles drift apart when they are randomly shuffled around? Of course the particles may be restricted by walls that they keep bumping into, or there may be some attractive force that makes them stick together, but let us assume that there are no such restrictions. The points move freely in a plane, only subject to the unpredictable force of a push in a random direction.

Suppose the point xn (at discrete time n) is perturbed by some stochastic vector ξ, defined in polar coordinates (r, α) with uniform density functions f, such that 

fr(ξr) = 1/R,  0 ≤ ξrR
fα(ξα) = 1/2π,  0 ≤ ξα < 2π.

Thus, xn+1 = xn + ξ, and the point may move to any other point within a circle centered around it and with radius R.

Now, suppose there is a point p which can move to any point inside a circle P in one step of time, and a point q that can move to any point within a circle Q.
First, suppose the point p remains at its position and the point q moves according to the probability density function. For the distance ||p-q|| to remain unchanged, q has to move to some point on the blue arc that marks points equidistant from p. As can be easily seen, the blue arc divides the circle Q in two unequal parts with the smallest part being closest to p. Therefore, the probability of q moving away from p is greater than the probability of approaching p. As the distance ||p-q|| increases, the arc through q obviously becomes flatter, thereby dividing Q more equally. In consequence, when p and q are close, they will be likely to move away from each other at a faster average rate than when they are farther apart, but they will always continue to drift apart.

After q has moved, the same reasoning can be applied to p. Furthermore, the same geometric argument works with several other probability density functions as well.

When a single point is repeatedly pushed around, it traces out a path and the result is a Brownian motion resulting in skeins such as this one.

Different probability density functions may produce tangles with other visual characteristics. The stochastic displacement vector itself may be Brownian noise, in which case the path is more likely to travel in more or less the same direction for several steps of time. Then two nearby points will separate even faster.

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