Olami-Feder-Christensen model
Encyclopedia
In physics
, in the area of dynamical systems, the Olami–Feder–Christensen model is an earthquake
model conjectured to be an example of self-organized criticality
where local exchange dynamics are not conservative. Despite the original claims of the authors and subsequent claims of other authors such as 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)
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.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, in the area of dynamical systems, the Olami–Feder–Christensen model is an earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...
model conjectured to be an example of self-organized criticality
Self-organized criticality
In physics, self-organized criticality is a property of dynamical systems which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune...
where local exchange dynamics are not conservative. Despite the original claims of the authors and subsequent claims of other authors such as 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
Square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group is known symbolically as p4m.Two...
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:-
-
else if the sites C1, C2, ..., Cm are critical the relaxation rule is applied in parallel:
-
-
where K'C is the tension prior to the relaxation and ΓC is the set of neightbours 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
-
this is equivalent to define a constant drive
-
and assume the relaxation step is instantaneous, which is a good approximation for an earthquake model.
Behaviour and criticality
The system's behaviour is heavily influenced by the α parameter. For α=0.25 the system is conservative (in the sense that the local exchange is conservative, as there is still tension loss in the borders) and clearly critical. For values α<0.25 the dynamics is very different, even in the limit α → 0.25, with greater noise and much greater transients. For low α, there are less possibilities of chain reactions which could lead to cut-offs in the earthquake size distribution, implying the model is not critical. Also, for α = 0, the model is trivially not critical.
These observations lead to the question of what is the value αc where the system makes the transition from critical to non-critical behaviour, which is still an open question.
-
-
-
-
-