| Summary: | Certain driven systems consisting of a large number of elements evolve towards a critical
state with no characteristic time or length scales. This class of phenomena is described
as Self-Organized Criticality (SOC). SOC relies on the condition of a slow driving of the
systems and the existence of fast burst-like responses of them and includes earthquakes.
We employ a model proposed by Xu et al. for ruptures in an elastic medium, subject
to shear stress, and apply it to the study of earthquakes. In the model, the size of an
earthquake is defined as the number of ruptures occurring sequentially on the basic units
of discretization (squares) of the medium. A histogram of the earthquake sizes shows
that the model is not completely scale invariant due to finite-size effects. To take them
into account, we implement a finite-size-scaling analysis. The results of this analysis show
that the model is scale invariant only when there is stress conservation. So, the model
displays SOC in the conservative case only.
We also study the dynamic quantities of the model, in particular the average stress
in the system. The sets of average stress values are analyzed using two types of time series
analysis. The nonlinear forecasting analysis investigates whether time series exhibit
low-dimensional chaotic behavior as opposed to high-dimensional (or stochastic) behavior.
We find that the above time series have a nonlinear structure, but with a substantial
stochastic component, so SOC is inherently high-dimensional. The appearance of nonlinear
structure is due to the fact that the system stops following linearly the external drive
when it releases stress through earthquakes. The rescaled range analysis characterizes
the time correlations (or memory effects) in the time series. We find strong positive time
correlations in the above time series. Their presence is due to the nature of the driving
in the model. These memory effects are destroyed as soon as a large earthquake resets
the system.
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