# Homework Help: Particle distribution, Diffusion

1. Nov 18, 2016

### Selveste

1. The problem statement, all variables and given/known data
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 $t$?

2. Relevant 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.

3. The attempt at a solution

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

2. Nov 18, 2016

### Staff: Mentor

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?
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).