Particle distribution, Diffusion

[Selveste]

Homework Statement

An initial particle distribution n(r, t) is distributed along an infinite line along the $z$-axis in a coordinate system. The particle distribution is let go and spreads out from this line.
$a)$ How likely is it to find a particle on a circle with distance $r$ from the $z$-axis at the time $t$?
$b)$ What is the most likely distance $r$ from origo to find a particle at the time $t$?

Homework Equations

The diffusion equation is given by
$$\frac{\partial n}{\partial t} = D \nabla^2 n$$
where $\nabla^2$ is the laplace-operator, $D$ is the diffusion constant and $n$ is the particle density.

The Attempt at a Solution

[/B]
I take it by "line along the z-axis" they mean ON the z-axis(?).
a) I am not sure how to go about this. Would it involve a Fourier transform, or can it be done more easily? Any help on where/how to start would be appreciated.
b) The most likely distance from the z-axis would be zero, because of symmetry(?). So the distance from origo would be z.
Thanks.

 

[mfb]

I take it by "line along the z-axis" they mean ON the z-axis(?).

I interpreted it in the same way.

a) What is the distribution of an initial point-like source? How can you generalize this to a 1-dimensional source?

b) The most likely distance from the z-axis would be zero, because of symmetry(?). So the distance from origo would be z.

There is no symmetry you can use as distance cannot be negative and different distances have different differential volumes. The most likely point will be on the z-axis, but the most likely distance won't. The problem statement is confusing (is it translated?), as r seems to be the radial direction, but then it is the distance to the z-axis, not the distance to the origin (where the most likely value would be very messy to calculate).