Theory of Random Walk
Random walk is a statistical process by which particles or photons diffuse through a system as a result of repeated scatterings.
The simplest example of random walk is one-dimensional motion on a line. Assume a photon originates at the origin (x = 0), and moves to the left or right with equal probability. After traveling a distance called the mean free path, l, the photon encounters a scattering center that will scatter it left or right with equal probability. So its position at the time of the first scattering is x1 = 0 +/- l. Its mean position is <x1> = 0. But its mean square position is <x12> = <(0 +/- l)2> = l2.
At the time of second scattering, the photon has position x2 = x1 +/- l. Its mean position is still zero, but now its mean square position is <x22> = <(x1 +/- l)2> = <x12 +/- 2x1l + l2> = <x12> + l2 = 2l2. Continuing this process, we find <xN> = 0, and <xN+12> = <xN2> + l2. So, we find:
<xN2> = Nl2.
The root mean square position after N scatterings is <xN2>1/2 = N1/2l. So, after 100 scatterings, the photon has typically moved a distance 10l from the origin, and after 106 scatterings, typically a distance 103l.
This very important result can easily be generalized to scattering in two or three dimensions. Let's say that each scattering sends the photon in a completely random direction, independent of the direction from which it encountered the scattering center. The square of the mean distance travelled is given by the Pythagorean theorem: d2 = Dx2 + Dy2 + Dz2 = l2. But, since the three directions are equally likely, we have <Dx2> = <Dy2> = <Dz2> = l2/3.
Now, we have for the (N+1)th scattering, <xN+12> = <xN2> + l2/3, and likewise <yN+12> = <yN2> + l2/3 and <zN+12> = <zN2> + l2/3. . So, the mean square radial distance is given by <RN+12> = <(xN+1 + yN+1 + zN+1)2> = <xN+12> + <yN+12> + <zN+12> + 2<(xN+1)(yN+1)> + 2<(yN+1)(zN+1)> + 2<(zN+1)(xN+1)> = <xN2> + <yN2> + <zN2> + l2 = <RN2> + l2. Thus, we find the root mean square radial displacement after N scatterings:
<RN2>1/2 = N1/2 l,
and the typical path length taken by a photon that has traveled a typical radial distance DR = <RN2>1/2 is L = Nl = DR2/l.
This path length can be huge. For example, in the interior of the Sun, l = 1 cm (typically), while R = 7 x 1010 cm. That means that N = (R/l)2 = 4.9 x 1021. If a photon could travel freely from the center of the Sun to its photosphere, it would do so in a timescale R/c = (7 x 1010)/(3 x 1010) = 2.3 sec. But it can't. It must travel a path length L = Nl = 4.9 x 1021 cm, and that will take a timescale L/c = 4.9 x 1021/(3 x 1010) = 1.6 x 1011 sec = 5000 years. That's roughly how long it takes radiation to leak out of the Sun.