# 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

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

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

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

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