Among the large variety of sandpile models, does someone know if there is a name for a variation of the BTW model implementing these features:
- Grains/chips are initially deposited in the same cell (therefore the system expands from the given cell)
- The 2d lattice is infinite (at least big enough to let the system stabilize without requiring sinks)
- Initially the lattice is empty
- For the rest consider canonical BTW parameters (threshold 4, von Neumann neighborhood)
In particular I am interested in knowing the relation between the radius (r) and the number of chips in the system (n), i.e. obviously in a system like that r ∝ n for any given state of equilibrium.
Does anyone know a work (or has an idea) that could help? Any hint is really welcome.
PS. Just to warn you, my enthusiasm about the subject is much better than my maths skills.
Please let me try to clarify the question about the radius with a SOLVED case in ONE-dimensional lattice.
Imagine an empty one-dimensional lattice, such as:
- - - - - - - - - - - - - - - - - - - -
where we can only deposite chips in the site x:
- - - - - - - - - x - - - - - - - - -
that way if we only have a chip z(x)=1 we would have:
- - - - - - - - - 1 - - - - - - - - -
the transition rule reads as follows: when z(x)>=k (with k the number of neighbors) we trigger an avalanche such that z(x) ← z(x) - k, and then the k neighbors gain a chip z(x±1) ← z(x±1)+1.
That is, for z(x)=2:
- - - - - - - - - 2 - - - - - - - - -
it holds that z(x)>=2 (i.e. in a 1d lattice k=2) and thus an avalanche is triggered: z(x) ← z(x) - 2,, and, z(x±1) ← z(x±1)+1. This results in the following state of equilibrium.
- - - - - - - - 1 0 1 - - - - - - - -
Centering the system in the position x (where now z(x)=0), we can see that it has become a radius (r) r=1.
If we consider now the case for 4 chips:
- - - - - - - - - 4 - - - - - - - - -
that would trigger an avalanche resulting in:
- - - - - - - - 1 2 1 - - - - - - - -
Given that z(x)>=2 still holds, the system is in a non-equilibrium state and then the avalanche goes on:
- - - - - - - - 2 0 2 - - - - - - - -
which in not yet in equilibrium, so goes on:
- - - - - - - 1 0 2 0 1 - - - - - - -
and yet goes on until the following state of equilibrium:
- - - - - - - 1 1 0 1 1 - - - - - - -
For this case we can see that r=2 (i.e. the center is in x).
If we let n be the number of chips, k the number of neighbors, and r the radius, for the unidimensional case (with k=2) is easy to show that:
r = floor(n/k)
What I am trying to figure out is which is the equation of the radius (r) for a 2-dimensinal case (grid lattice) using a von Neumann neighborhood, i.e. k=4.
Thanks for your time!!
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