Earthquake model

One of the classical examples of power-law distributions may be found in geology. It is the Gutenberg-Richeter law, which relates the number of earthquakes to their magnitude. Mathematically this relation is expressed as \( \lg N = a - b M \). Here \( N \) is a number of earthquakes of certain magnitude \( M \) or stronger, \( b \) is empirically determined and depends on seismic activity of the region, while \( a = \lg N_0 \).

In this text we will briefly present self-organized criticality model, which reproduces the power-law distribution of earthquakes - Olami-Feder-Christensen model.

The definition of model

Lets say that we have a lattice of known dimensions, \( S \). Each cell in the lattice has internal parameter, which is called friction, \( K_{ij} \). As time goes by friction of each cell increases, until any cells reaches critical state, \( K_{ij} \geq 1 \). Critical cells distribute their friction to their neighboring cells and relax their state to zero. The critically may spread thus causing cascades, the total number of critical cells at a given time gives us the magnitude of earthquake.

Mathematically this may be written down as follows. If there are no critical cells, then

\begin{equation} K_{ij} \leftarrow K_{ij} + (1-K_{max}) , \quad t\leftarrow t + (1-K_{max}) , \quad K_{max} = \mathrm{max}_{(i,j)\in S} K_{ij} . \end{equation}

On the other hand if there some critical cells, \( C_i \), then the following procedure should be repeated over all neighborhoods of critical cells, \( S_{C_i} \):

\begin{equation} K_{j} \leftarrow K_{j} + \alpha K_{C_i} \;\forall j \in S_{C_i} , \quad K_{C_i} \leftarrow 0 . \end{equation}

This algorithm is implemented in our HTML5 applet below. We invite you to try it and observe the emergence of power-law distribution.

HTML5 applet