I'm having a little trouble understanding what the stationary diffusion equation does. As I understand it, the standard heat equation is $u_t = -k\Delta u$, where $k$ is for conductivity.
My intuitive understanding of this is that the heat equation is describing a process going towards equilibrium described by Laplace's equation. ie. the heat equation describes something headed towards a system described by Laplace's equation.
The diffusion equation is a little more general $u_t = -\nabla \cdot (g \nabla u)$ and I assume that it works in the same way the heat equation does, but with some function g in the mix. I don't understand what $g$ is doing in this equation, but what I'm really confused about is the stationary version of this. ie $f = -\nabla \cdot (g\nabla u)$. What does this describe? Is is an equilibrium state? If so, what does $g$ contribute? Otherwise it's just Poisson's equation, no?
I have heard something about $g$ being chosen discretely from a distribution, what does this refer to? Quick explanations or references (using only very elementary physics if any) are appreciated.
Apologies beforehand if this seems exceptionally naive.