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# Sandpile

 If you download the NetLogo application, this model is included. You can also Try running it in NetLogo Web

## WHAT IS IT?

The Bak–Tang–Wiesenfeld sandpile model demonstrates the concept of "self-organized criticality". It further demonstrates that complexity can emerge from simple rules and that a system can arrive at a critical state spontaneously rather than through the fine tuning of precise parameters.

## HOW IT WORKS

Imagine a table with sand on it. The surface of the table is a grid of squares. Each square can comfortably hold up to three grains of sand.

Now drop grains of sand on the table, one at a time. When a square reaches the overload threshold of four or more grains, all of the grains (not just the extras) are redistributed to the four neighboring squares. The neighbors may in turn become overloaded, triggering an "avalanche" of further redistributions.

Sand grains can fall off the edge of the table, helping ensure the avalanche eventually ends.

Real sand grains, of course, don't behave quite like this. You might prefer to imagine that each square is a bureaucrat's desk, with folders of work piling up. When a bureaucrat's desk fills up, she clears her desk by passing the folders to her neighbors.

## HOW TO USE IT

Press one of the setup buttons to clear and refill the table. You can start with random sand, or a uniform number of grains per square, using the setup uniform button and the grains-per-patch slider.

The color scheme in the view is inspired by a "traffic light" pattern:

red = 3 grains yellow = 2 grains green = 1 grain black = 0 grains

If the animate-avalanches? switch is on, overloaded patches are white.

Press go to start dropping sand. You can choose where to drop with drop-location. If the drop-location is set to "mouse-click", you can drop sand by clicking on the view (the go button needs to be active for that to work.)

If you start out with a uniform distribution of 0 or even 1, it might take a while before you see avalanches. If you want to speed up this process, uncheck "view updates" for a few seconds, and then check it again. This makes the model run faster, because NetLogo does not have to draw the model every tick.

When animate-avalanches? is on, you can watch each avalanche happening, and then when the avalanche is done, the areas touched by the avalanche flash white.

Push the speed slider to the right to get results faster.

If you press explore, hovering over a square with your mouse will show how big the avalanche would be if a grain was dropped on that square.

## THINGS TO NOTICE

The white flashes help you distinguish successive avalanches. They also give you an idea of how big each avalanche was.

Most avalanches are small. Occasionally a much larger one happens. How is it possible that adding one grain of sand at a time can cause so many squares to be affected?

Can you predict when a big avalanche is about to happen? What do you look for?

Leaving animate-avalanches? on lets you watch the pattern each avalanche makes. How would you describe the patterns you see?

Observe the Average grain count plot. What happens to the average height of sand over time?

Observe the Avalanche sizes and the Avalanche lifetimes plots. This histogram is on a log-log scale, which means both axes are logarithmic. What is the shape of the plots for a long run? You can use the clear size data button to throw away size data collected before the system reaches equilibrium.

## THINGS TO TRY

Try all the different combinations of initial setups and drop locations. How does what you see near the beginning of a run differ? After many ticks have passed, is the behavior still different?

Use an empty initial setup and drop sand grains in the center. This makes the system deterministic. What kind of patterns do you see? Is the sand pile symmetrical and if so, what type of symmetry is it displaying? Why does this happen? Each cell only knows about its neighbors, so how do opposite ends of the pile produce the same pattern if they can't see each other? What shape is the pile, and why is this? If each cell only adds sand to the cell above, below, left and right of it, shouldn't the resulting pile be cross-shaped too?

Select drop by mouse click. Can you find places where adding one grain will result in an avalanche? If you have a symmetrical pile, then add a few strategic random grains of sand, then continue adding sand to the center of the pile --- what happens to the pattern?

## EXTENDING THE MODEL

Try a larger threshold than 4.

Try including diagonal neighbors in the redistribution, too.

Try redistributing sand to neighbors randomly, rather than always one per neighbor.

This model exhibits characteristics commonly observed in complex natural systems, such as self-organized criticality, fractal geometry, 1/f noise, and power laws. These concepts are explained in more detail in Per Bak's book (see reference below). Add code to the model to measure these characteristics.

Try coloring each square based on how big the avalanche would be if you dropped another grain on it. To do this, make use of push-n and pop-n so that you can get back to distribution of grains before calculating the size of the avalanche.

## NETLOGO FEATURES

• In the world settings, wrapping at the world edges is turned off. Therefore the neighbors4 primitive sometimes returns only two or three patches.

• In order for the model to run fast, we need to avoid doing any operations that require iterating over all the patches. Avoiding that means keeping track of the set of patches currently involved in an avalanche. The key line of code is:

set active-patches patch-set [neighbors4] of overloaded-patches

• The same setup procedure is used to create a uniform initial setup or a random one. The difference is what task we pass it for each pass to run. See the Tasks section of the Programming Guide in the User Manual for more information on Tasks.

• To enable explore mode, this model makes use of a data structure from computer science called a "stack". A stack works just like a stack of papers in real life: you place (or "push") items onto the stack such that the top of the stack is always the item most recently pushed onto it. You may then "pop" items off the top of the stack, revealing their value and removing them from the stack. Hence, stacks are good for saving and restoring the value of a variable. To save a value, you push the variable's value onto the stack, and then set the variable to whatever new value you want. To restore, you pop the value off and set the variable back to that value.

Explore mode actually only ever needs one item on the stack. However, the stack may be used to save as many values as one wants. Hence, you could extend this model to allow people to explore further and further into the future, and then let them pop back to their original place.

## RELATED MODELS

• Sand
• Sandpile 3D (in NetLogo 3D)

## CREDITS AND REFERENCES

https://en.wikipedia.org/wiki/Abelian_sandpile_model

https://en.wikipedia.org/wiki/Self-organized_criticality

Bak, P. 1996. How nature works: the science of self-organized criticality. Copernicus, (Springer).

Bak, P., Tang, C., & Wiesenfeld, K. 1987. Self-organized criticality: An explanation of the 1/f noise. Physical Review Letters, 59(4), 381.

The bureaucrats-and-folders metaphor is due to Peter Grassberger.

## HOW TO CITE

If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.

For the model itself:

Please cite the NetLogo software as: