Slide sem ttulo

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Criticality in the Olami-Feder-Christensen
model Transients and Epicenters
Carmen P. C. Prado Universidade de São
Paulo (prado_at_if.usp.br)
VIII Latin American Workshop on Nonlinear
Phenomena LAWNP 03 - Salvador, Bahia, 2003
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Carmen P. C. Prado (USP - SP) Josué X. de
Carvalho (USP, pos-doc) Tiago P. Peixoto (USP,
PhD st) Osame Kinouchi (Rib. Preto, USP) Suani
T. R. Pinho (UFBa)
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• Introduction
• SOC Olami-Feder-Christensen model (OFC)
• History
• Recent developments
• Recent results
• Transients
• Epicenters (real earthquaques)
• Our results

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Self-organized criticality
Bak, Tang, Wisenfeld, PRL 59,1987/ PRA 38, 1988
Sand pile model

• 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

Drive h( i ) h( i ) 1 Relaxation
if s( i) h(i1) - h(i) s( i) s( i) -
2 s(i1) s(i1) 11 s(i-1)
s(i-1) 1
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Does real sand piles exhibits power-laws?
Chicagos Jaeger, Liu, Nagel, PRL 62
(89) Jaeger, Nagel, Science 255 (92) Bretz et
al, PRL 69 (92)
Different sizes time scales Held et al, PRL
65 (90) Roserdahl, Vekic, Kelly PRE 47 (93)
Rice piles (Oslo) Frette et al, Nature 379
(96) A. Malthe-Sørenssen, PRE (96)

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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
Two distinct time scales punctuated
equilibrium Slow movement of tectonic plates
(years) Fast earthquakes (seconds)
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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)
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Modelo Olami-Feder-Christensen (OFC)
If some site becomes active , that is, if F gt
Fth, the system relaxes
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The size distribution of avalanches obeys a
power-law, reproducing the Gutemberg-Richter law
SOC even in the non conservative regime
Simulation for lattices of sizes L 50,100 e
200. Conservative case ? 1/4
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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 ?

S. Lise, M. Paczuski, PRE 63, 2001, PRE 64, 2001
? 0 model is non-critical ?
0.25 model is critical at which
value of ? ?c the system changes its behavior
???
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• First large scale simulation (Grassberger, PRE 49
  (1994), Middleton Tang, PRL 74 (1995) )
• periodic boundary conditions stationary state
  is periodic
• cross over (? 0.18 )
• small ?, ordered, period L2 , dominated by
  small avalanches ( s1)
• large ?, still periodic but disordered state
• open boundary condition
• also a cross over
• small ? bulk is ordered in a periodic state,
  s1, but close to the boundaries there is
  disorder most of the epicenters and large
  avalanches are in the boundaries
• large ? the whole lattice is prevented from
  ordering and large avalanches are also triggered
  in the interior of the lattice

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Spatial correlation starts from the borders
Random initial configuration
Josué X. Carvalho
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Spatial correlation starts from the borders
After 2 x 105 avalanches
Josué X. Carvalho
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Spatial correlation starts from the borders
After 10 x 108 avalanches
Josué X. Carvalho
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• More recent work (B. Drossel, PRL 89, 2002)
• The power - law distribution of avalanche sizes
  results from a complex interplay of several
  phenomena (part of them already pointed out in
  earlier papers), including
• boundary driven synchronization and internal
  desynchronization,
• limited float-point precision,
• slow dynamics within the steady state
• the small size of synchronized regions
• In the ideal situation of infinite floating
  point precision and L ? ?, the avalanche size
  distribution is dominated by avalanches of size 1
  , with the weight of large avalanches decreasing
  to zero with increasing system size. The model is
  not critical.

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• For the lattice model with periodic b.c.
• The stationary state is always periodic with all
  avalanches being of size 1.
• The failure to observe this in previous works
  are due to the small level of desynchronization
  caused by limited float-point precision.

• For the lattice model with open b.c.
• The study was concentrates on small values of ?
  (0.10).
• In the limits L ? ?, infinite precision, also
  all avalanches are of size1 (all sites topples
  with F Fth).

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In 2003, Miller and Boulter found again evidences
of the existence of a cross-over
G. Miller, C.J. Boulter, PRE 67, 2003

• Cross over ?x associated with the probability
  of finding an avalanche with s gt 1 , lower
  bound for ?c (concentrate on 0.20 lt ? ? 0.25)
• 0.12 ? ?x ? 0.16
• the result was not influenced by increasing the
  precision
• above this cross over , if ? gt ?x lt Fscgt gt1
  ( 10 -28)

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However...

• Their results have shown a qualitatively
  different behavior for the conservative regime
  indicating that ? 0.25 separates two different
  types of behavior in the OFC model
• although ?x 0.14, ?c 0.25 since ?x ?
  ?c
• the extrapolation procedure is not correct, ?x
  0.25 (what also leads to ?c 0.25)
• They observed universal features in the non
  conserving regime

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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 ?(?).
S Lise, H.J. Jensen, PRL 84, 2001 S. T. R.
Pinho, C. P. C. Prado, Bras. J. of Phys. 33
(2003). S. T. R. Pinho, C. P. C. Prado and O.
Kinouchi, Physica A 257 (1998). M. Chabanol,
V. Hakin, PRE 56 (1997) H.M Bröker, P.
Grassberger, PRE 56 (1997)
Almost critical O. Kinouchi, C.P.C. Prado, PRE 59
(1999)
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Branching rate in OFC and R-OFC
One counts the number of supercritical
descendents generated when a site topples
Remains controversial alternative extrapolation
procedures Christensen et al, PRL 87 (2001) de
Carvalho and C.P.C. Prado, PRL 87 (2001)
J. X. de Carvalho, C. P. C. Prado, Phys. Rev.
Lett. 84 , 006, (2000).
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Miller and Boulter, PRE 66, (2002)

• Layer branching rate ? i ( ?, L),
• i 1, ... L/2 indicates the distance of the
  site from the boundary
• L 1000 (non-conservative), L700
  (conservative)
• ?c 0.25
• average avalanche sizes ?(?, L) 1 - 1/s(?,
  L).
• Control models (beginning of organization)

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The qualitative difference in the behavior of the
conservative and non-conservative regime was also
observed in other situations
S. 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!
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Transients
J. X. de Carvalho, C.P.C. Prado, Physica A, 321
(2003)
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Stationary state is identified by the mean value
of the energy per site After a transient, ltFi,jgt
fluctuates around a mean value
Conservative case

• The beginning of stationary state is clearly
  identified
• Transient time scales with L2

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conservative and non-conservative regimes display
qualitatively different behavior during
transient

• Large fluctuations
• Much longer transient, scales Lb, b gt 2 (in
  this case 4)
• Initial bump scales L2

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Detail of beginning...
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Dynamics of the epicenters

• S. Abe, N. Suzuki, cond-matt/0210289
• 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
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Free-scale network degree of the node
(connectivity) 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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Studied the OFC model in this context
0.240
Clear scaling
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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Qualitative diference between conservative and
non-conservative regimes
0.249
0..25
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L 300
L 200
We need a growing network ...
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Different cell sizes
L 400, 2 X 2
L 200, 1 X 1
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Distribution of connectivity
L 200, ? 0.25
L 200, ? 0.249
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Conclusions

• Robustness of OFC model to describe real
  earthquakes
• Dynamics of epicenters
• Many open questions