I find it hard to see the differences between Brownian motion and diffusion.
As I understand, both are represented by the diffusion equation – am I right? And if I'm not, how is Brownian motion described?
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Diffusion comes about as the result of Fick's law in continuous mediums.
Fick's law briefly: if you have a "stuff" (which can be any conserved quantity) and you allow two containers of that stuff together so that they can share it, then the flow of the stuff will be from the container with greater concentration to the container of less concentration, and the flow will be proportional to the concentration difference. Some examples: my stove works by creating a burner with a high concentration of thermal energy; the "concentration" of this energy is measured as a temperature, and so when I put a colder pot on the burner heat flows into the pot, proportional to the temperature difference. Another "stuff" is momentum in the $x$-direction, so when I am in a canoe on a lake and I put my paddle into a body of water, as I move it through in one direction, it starts to pull the water along with it in that direction, sharing momentum with the water; this is how I move forward by pushing the water backwards. To move faster, I need to put more speed into my paddle.
Now if you have a lot of particles collectively undergoing Brownian motion, they will naturally diffuse according to Fick's law: divide a fluid into conceptual boxes with a coordinate system that is flowing downstream with the fluid. Each box holds an amount of particles proportional to their concentration at that point, and assuming the directions are entirely randomized, if you look at two boxes side-by-side then the number crossing from A to B will be proportional to A's concentration, while the number crossing from B to A will be proportional to B's concentration, and so the net effect will be a flow from higher to lower concentration, proportional to that difference in concentration.
But also note that Fick's law does not directly answer why these things are randomly moving in the first place; diffusion just emerges as the consequence of the fact that they are moving. And diffusion makes sense for "stuffs" that are harder to conceptualize as particles, like momentum-in-the-x-direction, or energy.