Slide sem t

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Dynamics of epicenters in the Olami-Feder-Christe
nsen Model
Carmen P. C. Prado Universidade de São
Paulo (prado_at_if.usp.br)
Trends and Perspectives in Non-extensive
Statistical Mechanics 60th-birthday of C.
Tsallis Angra dos Reis, Rio de Janeiro, 2003
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Tiago P. Peixoto (USP, PhD st) Osame Kinouchi
(Rib. Preto, USP) Suani T. R. Pinho (UFBa) Josué
X. de Carvalho (USP, pos-doc)
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• Introduction
• Earthquakes, SOC and the Olami-Feder-Christensen
  model (OFC)
• Recent results on earthquake dynamics
• Epicenter distribution (real earthquakes)
• Epicenters in the OFC model (our results)

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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

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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) Sornette and Sornette, Europhys. Lett.
(1989) Ito and Matsuzaki, J. Geophys. Res. B
(1990) )
Exhibits universal power - laws
Gutemberg-Richter s law (energy) P(E) ? E -B
Omori s law (aftershocks and foreshocks) n(t)
t -A
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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...
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Burridge-Knopoff model (1967)
Olami et al, PRL68 (92) Christensen et al, PRA
46 (92)
?
k
i - 1 i i 1
friction
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Modelo Olami-Feder-Christensen (OFC)
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The size distribution of avalanches obeys a
power-law, reproducing the Gutemberg-Richter law
and Omoris Law
N( t ) t -?
Hergarten, H. J. Neugebauer, PRL 88,
2002 showed that the OFC model exhibits sequences
of foreshocks and aftershocks, consistent with
Omori s law, but only in the non-conservative
regime!
Simulation for lattices of sizes L 50,100 e
200. Conservative case ? 1/4
SOC even in the non conservative regime
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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
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• 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
???
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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 O. Kinouchi, C.P.C. Prado, PRE 59
(1999)
J. X. de Carvalho, C. P. C. Prado, Phys. Rev.
Lett. 84 , 006, (2000).
Almost critical
Remains controversial
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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
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• 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

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S. Abe, N. Suzuki, cond-matt / 0210289
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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
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We studied the OFC model in this context, to see
if it was able to predict also this behavior
Clear scaling
0.240
( 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
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The exponent ? that characterizes the power-law
behavior of P(k), for different values of ?
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The size of the cell does not affect the
connectivity distribution P(k) ...
L 400, 2 X 2
L 200, 1 X 1
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But surprisingly,
There is a qualitative diference between
conservative and non-conservative regimes !
0..25
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L 300
L 200
We need a growing network ...
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Distribution of connectivity
L 200, ? 0.25
L 200, ? 0.249
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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
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• 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
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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...