Imagine that there's a box container shown in the graph. (You may consider it as a 2-dimensional rectangle.)
A division plate is in the middle of the container.
A sand pile is on the division plate, falling from the slits between container wall and division plate.
The sand would keep falling until it stops.
The question is what's the shape of the red line shown in the graph? Is there a mathematical solution for the shape?
I came here from the closed version of this question in the Physics Stack Exchange, but I'd like to provide a solution based on the sandpile model (https://en.wikipedia.org/wiki/Abelian_sandpile_model). It is possibly too restrictive to be interesting, but I think it is a solution.
Basically, given a 2D grid of the bottom of the container, each grid point can hold some sand. In the sandpile model, the amount of sand held before collapse is equal to the number of neighbors - so here, that's two, except at the boundary - let's say at the boundary, only one grain redistributes to the neighbor. So what shape is that?
It might be easy to see, but it's a straight line. The sand will fill the sides, the extra will tumble to the middle, and when every bin has 1 grain, it saturates. That's not exactly "equilibrium", since one you drop another grain, that grain will bounce around between the bins, but the simulation stops.
I made a silly Google Spreadsheet to show this as well:
https://docs.google.com/spreadsheets/d/1TUTONI_Ka33wAxkMOaFXWic5a1qIxotyXsmcplueihY/edit?usp=sharing
It's pretty dumb, since it requires partially updating by hand, but anyway...I think this is a solution for the simplest set of assumptions. A better version would somehow allow for a better gradient between grid points then $1:1$ - like maybe sand can "only move downhill" so that bins like $2$ $2$ $1$ $\to$ $3$ $2$ $1$ $\to$ $4$ $2$ $1$ tumbles to $3$ $2$ $1$ $1$ or something. In that case, I think the answer is the curve would be the absolute value function, centered at the middle.