Olami–Feder–Christensen model

In physics, in the area of dynamical systems, the Olami–Feder–Christensen (OFC) model is an earthquake model conjectured to be an example of self-organized criticality where local exchange dynamics are not conservative. The model is named after Zeev Olami, Hans Jacob S. Feder, and Kim Christensen who proposed it in 1992.

Despite the original claims of the authors and subsequent claims of other authors such as Stefano Lise, whether or not the model is self organized critical remains an open question.

The system behaviour reproduces some empirical laws that earthquakes follow (such as the Gutenberg–Richter law and Omori's Law).

Model definition

The model is a simplification of the Burridge-Knopoff model, where the blocks move instantly to their balanced positions when submitted to a force greater than their friction.

Let S be a square lattice with L × L sites and let Kmn ≥ 0 be the tension at site (m,n). The sites with tension greater than 1 are called critical and go through a relaxation step where their tension spreads to their neighbours. Through analogy with the Burridge-Knopoff model, what is being simulated is a fault, where one of the lattice's dimensions is the flaw depth and the other one follows the flaw.

Model rules

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

K

max

=

max

(

i

,

j

)

∈

S

K

i

j

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

K

i

j

←

K

i

j

+

(

1

−

K

max

)

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

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

K

C

i

←

0

,

i

=

1

,

…

,

m

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

K

j

←

K

j

+

α

K

C

i

′

∀

j

∈

Γ

C

i

,

i

=

1

,

…

,

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 ΓC 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.