Density of particles outside a constant point source

[Felian]

Hi, I'm having a strangely hard time imagining how the distribution of particles would look around a constant point source in some scenarios. I came up wiith something that sounds reasonable to me, but would like a logic check.

The easy one for me to grasp is distribution when the source emits particles that just move in straight lines without any interference from anything else. The inverse square law appears to describe both the amount of particles that leave a certain area, as well as the density of those particles in that area.

But suppose the stuff emitted by the point source is of such a kind that it doesn't just leave straight away. Say the particles of the stuff move like they're on purely random walks. Or say the stuff only spreads through internal pressure and is slowed down by stuff that is already further out and pushes back. Or say the source is emitting sheep in an endless 3D field that already had a base density of sheep, and these sheep like to graze in areas with lower sheep densities (or move randomly).

I'm thinking any of these scenarios would cause the density of stuff to increase since the amount of stuff that leaves an area lags behind that amount created there. But that eventually, there would appear a balance and the density distribution would smooth out to something stable (on average).

So, the thought I'd like a logic check on:
It seems to me that the amount of stuff that is moved outwards would depend on the derivative of the density shape (denser areas push stuff outwards faster, but will be pushed back at by the areas around them).
While the inverse square law must still describes the actual amount of stuff that actually leaves at every point, more can't possibly leave since you're limited to the output of the source, and if less leaves, the density should increase.
Since the two must be equal in a stable system, it makes me think that the density shape must be the antiderivative of the inverse square relationship, making the density relate to 1/distance.

Is that correct? Does it depend on the actual mechanics of the stuff? Is a shape like that even something that happens in any real system? Isn't it sneakily 1/distance^2 anyway or even 1/distance^3?

 

[Orodruin]

You are looking for the continuity and diffusion equations, which are basically what you are describing.

The continuity equation states that
$$\frac{\partial u}{\partial t} + \nabla \cdot \overline{j} = \kappa$$
where u is the concentration, $\overline j$ the current density, and κ the source density of a substance (particles, sheep, thermal energy, whatever you are studying). It essentially states that for a given volume, the change in the contained substance plus the flux out of the volume is equal to the amount produced in the volume.

Now to describe what you are trying to look at, a flow from higher concentration to lower, consider Fick's law
$$\overline j = - D \nabla u$$
which essentially states that the current density is proportional to the gradient, pointing in the direction of lower concentrations. Inserting Fick's law into the continuity equation yields the diffusion equation
$$\partial_t u - D \nabla^2 u = \kappa$$
where I have now assumed D is a constant. The stationary solution to the diffusion equation will depend on your boundary conditions as well as the stationary source term. In your case you wanted a point source, so this would correspond to κ being a delta distribution.

 

[Felian]

Thank you!

I'm afraid I don't have enough skill to actually solve an equation like that, but it gives me plenty of places to start looking.

Now to describe what you are trying to look at, a flow from higher concentration to lower, consider Fick's law
$$\overline j = - D \nabla u$$
which essentially states that the current density is proportional to the gradient, pointing in the direction of lower concentrations.

I'm assuming what you mean here is that the flow is proportional to the gradient of current density, or am I not understanding this correctly?

 

[Orodruin]

The flow through a surface is the integral over the surface of the current density projected onto the surface normal. The current density is (according to Fick's law) proportional to the gradient of the concentration of the substance. Sorry for being vague in the wording.