Models

How does one test the idea of self-organised criticality? It is quite hopeless to solve the complete dynamical equations for the relevant systems and try to see if they do self-organise to the critical state. Rather one makes highly simplified models to test the idea. If the models display self-organised criticality and are robust to various changes in detail at the microscopic level, then one can take that as supporting the plausibility of the proposal. A simple model of landslides or avalanches is that of a two-dimensional sand-pile. On a grid, define the local slope at each site (square on the grid) by an integer $Z$. The simulation procedure is as follows. First choose a critical value $Z_c$ and populate the grid with a starting configuration with each site having a random value less than $Z_c$. The updating rules are then:

1. Choose a random site and increase Z by one (add sand).
2. If Z exceeds critical value, reduce Z by four units and redistribute one unit to each of four neighbours.
3. Check if Z exceeds the critical value at any of the neighbours and continue redistribution process until the avalanche stops.
4. Count the total number of "topplings" involved in that avalanche.
5. Go to step (1)

Thus a number of avalanches of different sizes (number of topplings) are generated. By plotting the number of avalanches against their size for a large lattice (sand that goes out of the boundary is lost) one finds a power law over a large range of the parameters. A power law is also obtained for the distribution of lifetimes of the avalanches. What is interesting is that adding a single grain (step 1) can initiate an avalanche of any size, from one that involves only a few grains to one that involves almost the whole pile. That is, the system shows scale-invariance. (Notice that at the critical state the output is not proportional to the input, that is, the system is highly nonlinear and gives rise to "non-obvious" effects) After an avalanche has reduced the slope everywhere to below the critical value, the slow adding of send again brings the pile to another critical state and more avalanches. That is, the system continually self-organises to the unstable critical state from which scale-free avalanches occur. Although the effect of adding each particular grain is almost impossible to guess, the statistical distribution of avalanches according to an approximate inverse power law implies that small avalanches are more frequent than the larger ones. Note that a crucial feature of the models, which is a reflection of the real systems, is that the external process that drives the system (the inflow), occurs much slower than the faster internal reorganization processes that result in dissipation. Many other models have been investigated showing self-organised criticality. It has thus been suggested that self-organised criticality might not only be the reason for the diverse power laws in nature but also the dynamical mechanism behind fractal geometry and one-over-f temporal phenomena (see chapter 2).