The Sandpile Model

Simple rules can lead to complex behavior. Simulations of the system demonstrate how the behavior can be different each time, but that there are generalizations about the pattern of behavior that can be made. These complex systems have simple rules but multiple possible outcomes, i.e. they aren't deterministic.

Dropping sand grains one at a time onto a pile is one example of the complex behavior that can arise from a very simple set of rules. The rules are that:

sand grains are added one at a time

if, anywhere on the sandpile, there are two grains right on top of each other, there is a good chance that this pile of grains will fall over.

I'm just going to sketch a few steps in the building of a sand pile. There are simulations of this process available on the internet.

1. pile of sand develops
2. new grain added to top
3. grain could fall either direction
4. it happens to fall to the right
5. and then further tumbles
6. and finally ends up

 

At step 3 it could have fallen to the left, causing a bigger avalanche.

3. it could fall either way
4 - alternate. it falls to the LEFT
5 - alternate. causing a larger cascade

In one case one grain of sand tumbled down the pile, and in the other case it caused a larger event.

In a sandpile buildup there are lots of little tumbles, more small avalanches and only a few large avalanches. This is because if there hasn't been an avalanche for a while the pile gets steeper and steeper until it causes a large event. This model and the explanation has been explored in great deal in other sources (for example Bak 1996).

For the purposes of this example, we are interested in the frequency of the events and how big they are. It turns out that avalanches that are about twice as big are half as frequent. If you plot the frequency of events (Y axis) vs. the size of the event (X axis) you would get a plot that looks like this:

If you use a log-log plot, by simply making each axis a log scale, it looks like this:

The log-log transformation works because we are dealing with constant ratios of change; if the size increases by a certain ratio, then the frequency decreases by a related fraction. It doesn't matter where you are on the graph, whether you are at the second, or 82nd most frequent event, the ratios hold. This is an example of a scale independent relationship.

What does this relationship tell us about this system

self-organized criticality

linkages between events

Other examples of scale-independent relationships

Landslides -

Earthquakes - Gutenberg Richter Law

Size of cities - Zipf's Law

others?

Comments on this type of science compared to deterministic

compared to the example on pollution input

empirical results analyzed for this relationship

a simple rule results in many different paths - not deterministic

Management implications

compared to the example on pollution input

descriptive but not predictive

self-organization might be able to be managed (for example growth of new cities), but the regulation of that would have to be very complex - Ashby's Law.

Management of these complex systems requires a new kind of leadership (Margaret Wheatley 1999).

Conclusion

These natural and biological systems show scale independence. There are ecological systems that show very specific scale dependence, with different processes occurring at particular scales and governing the overall system.