Solved – Probability distribution of the magnitude of a circular bivariate random variable

Diffusion of a single particle is a random walk in two dimensions. As a function of time, $t$, the probability density for its location will therefore be Gaussian, centered at the particle's original location, with a variance $sigma^2 t$ directly proportional to time. (The constant of proportionality $sigma^2$ is a quantitative expression of the diffusivity of the medium.)

Because diffusion is usually modeled as an additive process–the motions of the particles are assumed to be independent and non-interfering–a uniform distribution of particles within any region $mathcal R$ will evolve into a uniform location mixture of such Gaussians. In statistical terminology, the distribution at time $t$ will be the sum of the uniform distribution over $mathcal R$ and the (two-dimensional) Gaussian distribution centered at $(0,0)$ with covariance matrix

$$Sigma(t) = pmatrix{sigma^2 t & 0 \ 0 & sigma^2 t}.$$

Because variances of independent variables add, the variance of the distribution at time $t$ will be the variance of the uniform distribution over $R$, plus $Sigma(t)$.

It is straightforward to compute the variance of a uniform distribution over an annulus of radii $r_1 lt r_2$: it is the diagonal matrix with values $(r_1^2 + r_2^2)/4$. Therefore the variance of the diffused particles after time $t$ is given by the matrix

$$left(frac{r_1^2 + r_2^2}{4}+ sigma^2 tright)pmatrix{1 & 0 \ 0 & 1}.$$

As $t$ grows large, the distribution becomes closer and closer to the solution obtained by starting with all the points near the centroid of $mathcal R$–regardless of what shape $mathcal R$ might have or of how the particles were distributed within it. Provided $mathcal R$ is bounded–as in this case–the diffused particles will eventually adopt a (circularly symmetric) Gaussian distribution centered at the centroid of $mathcal R$. Its variance is given by the preceding formula.

To illustrate, I placed 10,000 points uniformly and randomly within an annulus of radii $7/4$ and $2$ and diffused them via a random walk with Gaussian steps of variance $sigma^2 = 1/8^2$.

Along the way I kept track of the mean variance of the $x$ and $y$ coordinates relative to the theoretical variance as given above: namely, the starting variance plus $sigma^2 t$. Their ratio will not be constant, due to the random fluctuations: it is a form of a random walk itself. But it will stay close to $1$:

The progress of the diffusion can be plotted by showing the (smoothed) density on a 3D plot (whose height depicts the density).

Notice that by time $1/sigma^2 = 64$, the average amount of particle motion is sufficient to place many of them in the very center. The annulus appears to be "filled up", and thereafter the mode of the distribution is in the center of the ring. This filling is apparent in a left-to-right scan of these images, which will take you from step 1 through step 72 in a quadratic progression of intervals. (The image will show detail when magnified.)