Transcript constantino - Universidade de São Paulo

Dynamics of epicenters in the
Olami-Feder-Christensen Model
Carmen P. C. Prado
Universidade de São Paulo
([email protected])
Trends and Perspectives in Non-extensive Statistical Mechanics
60th-birthday of C. Tsallis
Angra dos Reis, Rio de Janeiro, 2003
Tiago P. Peixoto (USP, PhD st)
Osame Kinouchi (Rib. Preto, USP)
Suani T. R. Pinho (UFBa)
Josué X. de Carvalho (USP, pos-doc)
Introduction:
• Earthquakes, SOC and the Olami-FederChristensen model (OFC)
Recent results on earthquake dynamics:
• Epicenter distribution (real earthquakes)
• Epicenters in the OFC model (our results)
Self-organized criticality
“Punctuated equilibrium”
Extended systems that, under
some slow external drive
(instead of evolving in a slow and continuous way)
• Remain static (equilibrium) for
long periods;
• That are “punctuated” by very
fast events that leads the systems
to another “equilibrium” state;
• Statistics of those fast events
shows power-laws indicating
criticality
Bak, Tang, Wisenfeld, PRL
59,1987/ PRA 38, 1988
Sand pile model
Earthquake dynamics is probably the best “experimental ”
realization of SOC ideas ...
The relationship between SOC concepts and the dynamics of earthquakes
was pointed out from the beginning
(Bak and Tang, J. Geophys. Res. B (1989);
Matsuzaki, J. Geophys. Res. B (1990) )
Sornette and Sornette, Europhys. Lett. (1989); Ito and
Exhibits universal power - laws
Gutemberg-Richter ’s law (energy)
P(E)  E
-B
Two distinct time scales &
punctuated equilibrium
Slow: movement of
tectonic plates (years)
Omori ’s law (aftershocks and foreshocks) Fast: earthquakes (seconds)
n(t) ~ t -A
By the 20 ies scientists
already knew that most of the
earthquakes occurred in
definite and narrow regions,
where different tectonic
plates meet each other...
Burridge-Knopoff model (1967)
Moving plate
Olami et al, PRL68 (92); Christensen
et al, PRA 46 (92)
Fixed plate
V

k
i-1
i
i+1
friction
Olami et al, PRL 68, (1992); Christensen et al, Phys. Rev. A 46, (1992).
Perturbation:
F (i, j )  F (i, j )  
If some site becomes “active” , that is, if F > Fth, the system relaxes:
Relaxation:
F (i, j )  0
k
 
4k  
a
1
4
 0
F (i  1, j  1)  F (i  1, j  1)   F (i, j )
(i-1,j)
(i,j)
If any of the 4 neighbors exceeds Fth, the relaxation rule
is repeated.
 Fij
(i,j-1)
(i,j+1)
(i+1,j)
This process goes on until F < Fth again for all sites of
the lattice
The size distribution of avalanches obeys a power-law, reproducing
the Gutemberg-Richter law and Omori’s Law
N( t ) ~ t -
Hergarten, H. J. Neugebauer,
PRL
88, 2002
Simulation for lattices of sizes L = 50,100 e 200.
Conservative case:  = 1/4
SOC even in the non
conservative regime
showed that the OFC model exhibits
sequences of foreshocks and
aftershocks, consistent with Omori’
s law,
but only in the non-conservative
regime!
While there are almost no doubts about the efficiency of this
model to describe real earthquakes,
the precise behavior of the model in the non conservative
regime has raised a lot of controversy, both from a numerical
or a theoretical approach.
The nature of its critical behavior is still not clear. The model
shows many interesting features, and has been one of the
most studied SOC models
• First simulations where performed in very small lattices ( L ~ 15 to 50 )
• No clear universality class P(s) ~ s- ,  =  ( )
• No simple FSS, scaling of the cutoff
• High sensibility to small changes in the rules (boundaries, randomness)
• Theoretical arguments, connections with branching process,
absence of criticality in the non conservative random neighbor version
of the model has suggested conservation as an essential ingredient.
• Where is the cross-over ?
=0
model is non-critical
 = 0.25
model is critical
at which value of  = c the system
changes its behavior ???
Branching rate approach
Most of the analytical progress on the RN -OFC used a formalism developed by
Lise & Jensen which uses the branching rate ().
Almost critical
J. X. de Carvalho, C. P. C. Prado,
O. Kinouchi, C.P.C. Prado, PRE 59 (1999)
Phys. Rev. Lett. 84 , 006, (2000).
c
Almost critical
Remains controversial
Dynamics of the epicenters
S. Abe, N. Suzuki, cond-matt / 0210289
• Instead of the spatial distribution (that
is fractal) , the looked at the time
evolution of epicenters
•Found a new scaling law for
earthquakes (Japan and South
California)
Fractal distribution
S. Abe, N. Suzuki, cond-matt / 0210289
Time sequence of epicenters from
earthquake data of a district of
southern California and Japan
• area was divided into small cubic
cells, each of which is regarded as
vertex of a graph if an epicenter
occurs in it;
• the seismic data was mapped into
na evolving random graph;
Free-scale behavior of Barabási-Albert type
S. Abe, N. Suzuki, cond-matt / 0210289
Free-scale network
connectivity of the node
P(k) ~ k -
Complex networks describe
a wide range of systems in
nature and society
R. Albert, A-L. Barabási,
Rev. Mod. Phys. 74 (2002)
Random graph
distribution is Poisson
We studied the OFC model in this context, to see if it
was able to predict also this behavior
0.240
Clear scaling
( Curves were shifted upwards
for the sake of clarity )
0.249
Tiago P. Peixoto, C. P. C. Prado, 2003
L = 200, transients of 10 7, statistics of 10 5
The exponent  that characterizes the power-law
behavior of P(k), for different values of 
The size of the cell does not affect the connectivity
distribution P(k) ...
L = 400,
2X2
L = 200,
1X1
But surprisingly,
There is a qualitative diference between
conservative and non-conservative regimes !
0..25
L = 300
L = 200
We need a growing network ...
Distribution of connectivity
L = 200,  = 0.25
L = 200,  = 0.249
Spatial distribution of
connectivity, (non-conservative)
(b) is a blow up of (a);
The 20 sites closer to the
boundaries have not been plotted
and the scale has been changed
in order to show the details.
It is not a boundary effect
Spatial distribution of
connectivity, (conservative)
• In (a) we use the same scale of
the previous case
• In (b) The scale has been
changed to show the details of the
structure
Much more homogeneous
Conclusions
• Robustness of OFC model to describe real earthquakes, since its
able to reproduce the scale free network observed in real data
• New dynamical mechanism to generate a free-scale network, The
preferential attachment present in the network is not a rule but a
signature of the dynamics
• Indicates (in agreement with many previous works) qualitatively
different behavior between conservative and non-conservative
models
• Many open questions...