Model rules

If there are no critical sites, then the system suffers a continuous drive, until a site becomes critical:

${\displaystyle K_{\max }={\underset {(i,j)\in S}{\max }}K_{ij}\,}$

${\displaystyle K_{ij}\leftarrow K_{ij}+(1-K_{\max })\,}$

else if the sites C₁, C₂, ..., C_m are critical the relaxation rule is applied in parallel:

${\displaystyle K_{C_{i}}\leftarrow 0,\quad i=1,\ldots ,m\,}$

${\displaystyle K_{j}\leftarrow K_{j}+\alpha K'_{C_{i}}\,\forall \,j\in \Gamma _{C_{i}},\quad i=1,\ldots ,m}$

where K'_C is the tension prior to the relaxation and Γ_C is the set of neighbours of site C. α is called the conservative parameter and can range from 0 to 0.25 in a square lattice. This can create a chain reaction which is interpreted as an earthquake.

These rules allow us to define a time variable that is update during the driving step

${\displaystyle t\leftarrow t+(1-K_{\max })\,}$

this is equivalent to define a constant drive

${\displaystyle {\frac {dK_{i}}{dt}}=1\,\forall \,i\in S}$

and assume the relaxation step is instantaneous, which is a good approximation for an earthquake model.