terça-feira, 30 de novembro de 2010

Simulation on Sand Avalanches

It is common to observe how complex systems in nature present similar patterns, what is due to the self-organized criticality. Critical points work as attractors towards which systems evolve seeking for equilibrium. A common example are the sand piles. It is observed that as more grain of sand are dropped, new avalanches may occur. The avalanches may be of different proportions, local small avalanches or big ones, spreading widely in the pile. The observation of these avalanches shows that they follow a Zipf's law.

Here I present the results of a simulation of sand avalanches. The video shows a set of samples, which with 10 continuous seconds taken every 80 seconds from the simulation. At every second a new sand is randomly dropped on the board. If a certain peak has 4 more grain of sand then any of its neighbours, an avalanche happens towards this neighbour, what might lead to a cascade of avalanches.

The histogram bellow show the frequency of occurrence of avalanches of different sizes, described as: 0 (no avalanche), 1 (avalanche happened just on the level the grain was dropped), 2 (avalanche spreads to the next level), and so forth.

If we present the result above as a log-log plot of the types of avalanches (according to their magnitude) ranked according to their frequency of occurrence, we get to the figure bellows, which clearly is a power law, a Zipf's law. The type of avalanche with rank one is the most frequent, the type ranked two is the second most frequent, and so on. The plot bellow shows the rank vs. frequency of occurrence. The numbers presented near the curve (1 2 3 0 4 5 6 ...) are the type of avalanche: 1 stands for avalanche only in the level where the grain is dropped; 2 when the avalanche spreads to the next level; 3 when it spread two levels bellow; 0 when there is no avalanche; etc.

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