# Random walks and diffusion limits

Imagine a long and narrow cylinder of radius r and a point particle that moves in the region bounded by the cylinder. The motion is specified as follows: starting at a point on the inner wall of the cylinder, choose at random a direction and let the particle move with constant speed until it hits another point of the cylinder. Once there, choose a new direction at random and repeat the process. The problem is to determine the probability that the particle will be given distance away from the initial point at a given time in the future.

I realized that t is hard to find such a probability explicitly, but if the cylinder is very narrow and the particle moves very fast (with speed proportional to the reciprocal of the radius) you can use the central limit theorem to obtain an explicit (Gaussian) approximation. What is the variance of the resulting normal law? How does the variance change if the cross section of the tube is, say a square, instead of a circle?

Can Anyone help me here?

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I believe that polar coordinates and some calculus are used for this problem. –  NasuSama Jan 3 at 2:49

Not sure the CLT applies at all...

In the analogous dynamics in dimension $2$, the particle moves in the plane $(x,y)$ bouncing back and forth between the walls $y=0$ and $y=1$, choosing an angle $t$ in $(0,\pi)$ uniformly at random and getting a displacement $x=\cot t$. This implies that $P[|x|\geqslant u]\sim 2/(\pi u)$ when $u\to\infty$, hence $|x|$ is not integrable.

In dimension $3$, assume without loss of generality that the cylinder has equation $y^2+z^2=z$ in the coordinate system $(x,y,z)$ (thus, $y=z=0$ is a line on the surface of the cylinder and the diameter is $1$). Starting from $(0,0,0)$, the particle moves along the line $x=r\cos t\cos s$, $y=r\cos t\sin s$, $z=r\sin t$, where $t$ and $s$ are independent and uniform on $(0,\pi)$. The length of the displacement until the particle hits the cylinder again is $r=\sin t/(\sin^2t+\cos^2t\sin^2s)$, for a distance along the $x$-axis of $|x|=r\cos t\cos s=\sin t\cos t\cos s/(\sin^2t+\cos^2t\sin^2s)$.

Now, $|x|$ is large when $(t,s)\to(0,0)$, and then $|x|\sim t/(t^2+s^2)$. Using polar coordinates for $(t,s)$, that is, introducing $(\varrho,\alpha)$ such that $t=\varrho\cos\alpha$, $s=\varrho\sin\alpha$, one gets $|x|\sim \cos\alpha/\varrho$. Thus, $P[|x|\geqslant u\mid\alpha]\sim C\int\limits_0^{\cos\alpha/u}\varrho\mathrm d\varrho=C\cos^2\alpha/u^2$ and $P[x\geqslant u]\sim C/u^2$ when $u\to\infty$, where the various occurrences of $C$ are absolute constants whose value can vary from line to line. In particular, $|x|$ is not square integrable.

It seems that for a cylinder in $\mathbb R^{d+1}$ whose section is a ball in $\mathbb R^d$, the displacement $|x|$ in one step is such that $E[|x|^\nu]$ is finite if and only if $\nu\lt d$. In particular one would expect CLT for cylinders in dimension at least $d+1=4$.

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Choosing the two angles $t$ and $s$ to be uniform on $(0,\pi)$ doesn't give a uniformly random direction, does it? –  mjqxxxx Jan 1 at 3:22
No it doesn't, since the surface element is proportional to $\cos t\mathrm ds\mathrm dt$. –  Did Jan 1 at 15:34