BTW Sandpile Model

One of the things most of us loved to do in the beach as children (or maybe even as adults?) was to play with sand. We put grains of sand on top of each other and sometimes get frustrated when the pile topples. 

So, why do the grains of sand to topple? And when do they topple?

Initially, we have a random distribution of sand over a finite space. Adding more grains of sand causes the slope of the pile to become steeper. Some sites become more unstable than others such that they can no longer hold any more grains and any additional grain will simply tumble down the pile. Sometimes, the pile reaches a state in which a single grain that is tumbling down may carry others with it, causing a large-scale avalanche. We may technically call this a critical state of the sandpile. Seeing as the criticality arose without tuning any parameters, this phenomenon is called self-organized criticality (SOC), a concept which was first suggested by Per Bak, Chao Tang and Kurt Wiesenfeld in their 1987 paper [1]. The concept of SOC is now widely applied across different fields.

In the same paper they introduced the Bak-Tang-Wiesenfeld (BTW) sandpile model which is one of the dynamical systems that exhibited SOC. The BTW model is also an example of a 2D cellular automaton.

Algorithm

In this activity, we implemented the BTW sandpile model using the following algorithm:

1. Initialize random $N \times N$ array $A$ with possible states $S = {0, 1, 2, 3}$.
2. Iteration{
• Randomly select a site (i, j) and update the state of the cell by adding 1, i.e. $S_{t+1} = S_{t} + 1$.
• Check whether the state of the cell (i, j) satisfies $A_{ij} \ge 4$
• i. If Yes: Then the site is unstable and it can topple its "contents" to its four neighbors (i.e. Von Neumann neighborhood) so you update the values of the cells to:
• $A_{ij} \longrightarrow A_{ij} - 4$
• $A_{i-1,j} \longrightarrow A_{i-1,j}  + 1$
• $A_{i+1,j}\longrightarrow  A_{i+1,j}  + 1$
• $A_{i,j+1} \longrightarrow  A_{i,j +1} + 1$
• $A_{i,j-1} \longrightarrow  A_{i,j -1} + 1$

• If No: Proceed to next iteration
• Check the neighborhood for other unstable sites and repeat the toppling process until all sites are stable before proceeding to the next iteration. 
• Note: There are sinks along boundaries of the grid so for cells along the edge that topple  the grains are "thrown" in the sinks.}

In our implementation, a 128 x 128 array was used and the system was allowed to evolve for T = 100, 000 iterations. In each iteration, we take note of the the avalanche size or the number of cells affected by the the toppling due to a cell's instability.

Results

The time series for the avalanche sizes is shown in Figure 2. It can be observed that at around t ~30,000, the system enters its critical state. The reaction of the system to small perturbations becomes highly nonlinear, resulting to high uncertainties in the outcome of the the disturbance. The effect of the perturbation ranges from small shifts to a global avalanche depending on where the perturbation is applied. In the sandpile analogy, addition of a "grain of sand" can either cause a global avalanche or small rearrangements in the distribution of sand in localized regions.  This is a manifestation of the system's self-organized criticality.

Figure 2. Time series for the avalanche sizes

The animations below show the behavior of the system before and after it achieved its critical state. These images show the sandpile before the cells are updated in case of an avalanche. The images show that when the system reaches its critical state the sandpile become dense.

(a) Before reaching critical state

(b) After reaching critical state

Figure 3. Animations showing the bahavior of the system (a) before and (b) after reaching the critical state 

We also examine the distribution of the avalanche sizes, shown in Figure 3 in logarithmic scale. The linear trend suggests a power-law size distribution which is characteristic of SOC. From the fit, the power law exponent of the system is $\alpha = -1.195$.

Figure 4. Distribution of avalanche sizes

References:
1. Bak, Per, Chao Tang, and Kurt Wiesenfeld: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett., 59(4):381–384, Jul 1987.
2. S. Lübeck and K. D. Usadel., Bak-Tang-Wiesenfeld sandpile model around the upper critical dimension.  Phys. Rev. E 56, 5138 (1997).


Collaborators:

Mary Angelie M. Alagao

Maria Eloisa M. Ventura