a)

This is the behaviour I expect from both concentrations at t= 0. Based on the theory of diffusion I would say that as t goes on, the Gaussian distributions will spread out and after a large number of collisions both concentrations will end up spreading out more or less evenly throughout the whole volume. Please let me know if this reasoning is wrong.

A doubt came to my mind as I imagine the same scenario if we had absorbing boundaries instead of reflecting ones. Therefore, does really matter the kind of boundary we are dealing with when we are trying to predict how the two concentrations (in this case) are going to spread out as time flows?

EDITED:

My initial idea about the identical scenario regarding either absorbing boundaries or reflecting ones was wrong. That is because after a long time in presence of absorbing boundaries, the concentration will end up being zero.

Related question: http://physics.qandaexchange.com/?qa=3113/solving-the-diffusion-equation-with-an-absorbing-boundary

b)

Based on what I have said before, I would expect the limiting case to be that both concentrations were equal to each other.

EDITED:

Actually I was wrong as there is only one distribution of concentration, which tends to be uniform after a long time (as explained at a)).

**I have been looking for a deeper explanation and I came up with the idea that at the boundaries the current is zero:**

$$j =-D\frac{\partial c(+-a/2,t)}{\partial x} = 0$$

**Which means that c(x,t) has vanishing derivatives. How could this fact help to explain the limiting behaviour of the concentration after a long time?**

c)

This household potential:

As we are dealing with the distribution of particles in thermal equilibrium, the first thing that came to mind when I read potential was the Boltzmann equation:

$$n = ke^{-V(x)/KT}$$

EDITED:

**I am coming up with the Boltzmann distribution because it can tell us the distribution of molecules. If the potential energy is known as a function of distance, then the proportion of them at different distances is given by this law. And we do know the potential, which is the square-well. So in the equilibrium concentration I should be able to check whether with the square-well potential we get:
- When $|x| \le \frac{a}{2}$ we get a constant (the value of the concentration in equilibrium between the boundaries).
- When $|x| > \frac{a}{2}$ we get zero (outside the barriers).
But how can I check this?**

d)

**I came across an explanation. It is not detailed and I barely understand it. Please explain it if you see that it makes sense:** (I do not know why pasteboard did not work. I had to use Imgur):

a) It is clear.

b) Please read bold text.

c) Please read bold text.

d) Please read bold text.