Relationship between density and probability in diffusion

[It's me]

Homework Statement

Consider the diffusion of a drop of ink in a water vase. The density of the ink is $\rho (\vec{r}, t)$, and the probability $P(\vec{r}, t)$ obeys the diffusion equation. What is the relationship between $\rho (\vec{r}, t)$ and $P(\vec{r}, t)$?

Homework Equations

The Attempt at a Solution

I think the probability is proportional to the density, but I am not sure in what way. I don't understand how to approach this problem.

 

[haruspex]

The probability of what?

 

[It's me]

The probability of a particle for being at a position $\vec{r}$ at time $t$.

 

[haruspex]

The probability of a particle for being at a position $\vec{r}$ at time $t$.

What does it mean to say that there is a particle of dye at a precise point in space? Is there some preamble to the question, telling you to consider the volume as made up of little boxes, each of which contains either one particle of dye or none?

 

[It's me]

Not really, it just says that the probability is that of Brownian movement and we are asked to find that relationship as preparation to derive and solve the diffusion equation.

 

[haruspex]

the probability is that of Brownian movement

I still don't think that means anything. How about, it's the probability density function in a Brownian movement?

 

[It's me]

Okay yes that makes sense.