Criticality in the Olami-Feder-Christensen model:Criticality in the Olami-Feder-Christensen model:

Transients and EpicentersTransients and Epicenters

VIII Latin American Workshop on Nonlinear PhenomenaVIII Latin American Workshop on Nonlinear Phenomena LAWNP ’03LAWNP ’03 - - Salvador, Bahia, 2003Salvador, Bahia, 2003

Carmen P. C. Prado Universidade de São Paulo

(prado@if.usp.br)

Carmen P. C. Prado (USP - SP)Carmen P. C. Prado (USP - SP)

Josué X. de Carvalho (USP, pos-doc) Josué X. de Carvalho (USP, pos-doc)

Tiago P. Peixoto (USP, PhD st)Tiago P. Peixoto (USP, PhD st)

Osame Kinouchi (Rib. Preto, USP)Osame Kinouchi (Rib. Preto, USP)

Suani T. R. Pinho (UFBaSuani T. R. Pinho (UFBa))

IntroductionIntroduction• SOC & Olami-Feder-Christensen model (OFC)

• History

•Recent developments

Recent resultsRecent results•Transients

• Epicenters (real earthquaques)

•Our results

Self-organized criticalitySelf-organized criticality

““Punctuated equilibrium”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

Drive h( i ) h( i ) + 1

Relaxation if s( i) = h(i+1) - h(i)

s( i) s( i) - 2

s(i+1) s(i+1) +11

s(i-1) s(i-1) + 1

Chicago’s: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)

Does real sand piles exhibits power-laws?Does real sand piles exhibits power-laws?

Earthquake dynamics is probably the best “experimental ” realization of SOC ideas ...

Exhibits universal power - lawsExhibits universal power - laws

Gutemberg-Richter ’s lawGutemberg-Richter ’s law (energy)

P(E) E -B

Omori ’s lawOmori ’s law (aftershocks and foreshocks)

n(t) ~ t -A

Two distinct time scalesTwo distinct time scales & & punctuated equilibriumpunctuated equilibrium

Slow:Slow: movement of tectonic plates (years)

Fast:Fast: earthquakes (seconds)

The relationship between SOC concepts and the dynamics of earthquakes The relationship between SOC concepts and the dynamics of earthquakes was pointed out from the beginningwas 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) )

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)Burridge-Knopoff model (1967)

Fixed plate

Moving plate

V

k

i - 1 i i + 1 atrito

thij FFIf

ijijij

ij

FFF

Fthen

11

0

k

k

4

4

1a 0

Olami et al, PRL68 (92); Christensen et al, PRA 46 (92)

Perturbation:Perturbation: ),(),( jiFjiF

If some site becomes “active” , that is, if F > Fth, the system relaxes:

Relaxation:Relaxation: 0),( jiF

),()1,1()1,1( jiFjiFjiF

(i,j)

(i-1,j)

(i+1,j)

(i,j-1) (i,j+1)

Fij

If any of the 4 neighbors exceeds Fth, the relaxation rule is repeated.

This process goes on until F < Fth again for all sites of the lattice

Olami et al, PRL 68, (1992); Christensen et al, Phys. Rev. A 46, (1992).

k

k

4 4

1a 0

The size distribution of avalanches obeys a power-law, reproducing The size distribution of avalanches obeys a power-law, reproducing

the Gutemberg-Richter lawthe Gutemberg-Richter law

Simulation for lattices of sizes L = 50,100 e 200.Conservative case: = 1/4

SOC even in the non conservative regime

While there are almost no doubts about the efficiency of this While there are almost no doubts about the efficiency of this model to describe real earthquakes, model to describe real earthquakes,

the precise behavior of the model in the non conservative the precise behavior of the model in the non conservative regime has raised a lot of controversy, both from a numerical regime has raised a lot of controversy, both from a numerical

or a theoretical approach.or a theoretical approach.

The nature of its critical behavior is still not clear. The model The nature of its critical behavior is still not clear. The model shows many interesting features, and has been one of the shows many interesting features, and has been one of the

most studied SOC modelsmost 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 criticalat which value of = c the system changes its behavior ???

S. Lise, M. Paczuski, PRE 63, 2001, PRE 64, 2001

First large scale simulation (Grassberger, PRE 49 (1994), Middleton & Tang, PRL 74 (1995) )

• periodic boundary conditions: stationary state is periodicperiodic boundary conditions: stationary state is periodic

• cross over ( ~ 0.18 ):

• small , ordered, period = L2 , dominated by small avalanches ( s=1)

• large , still periodic but disordered state

• open boundary condition:open boundary condition:

• also a cross over

• small : bulk is ordered in a “periodic” state, s=1, 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;

Josué X. Carvalho

Spatial correlation starts from the bordersSpatial correlation starts from the borders

Random initial configuration

25.00.0),( jiF

50.025.0),( jiF

75.050.0),( jiF

00.175.0),( jiF

Josué X. Carvalho

After 2 x 105 avalanches

25.00.0),( jiF

50.025.0),( jiF

75.050.0),( jiF

00.175.0),( jiF

Spatial correlation starts from the bordersSpatial correlation starts from the borders

Josué X. Carvalho

After 10 x 108 avalanches

25.00.0),( jiF

50.025.0),( jiF

75.050.0),( jiF

00.175.0),( jiF

Spatial correlation starts from the bordersSpatial correlation starts from the borders

More recent workMore 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,limited float-point precision, • slow dynamics within the steady state • the small size of synchronized regions;

• In the ideal situation of infinite floating point precisioninfinite floating point precision and L 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.

For the lattice model with periodic b.c.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.For the lattice model with open b.c.

• The study was concentrates on small values of (~0.10).

• In the limits L L , infinite precisioninfinite precision, also all avalanches are of size1 (all sites topples with F = Fth).

In 2003, Miller and Boulter found again evidences of the In 2003, Miller and Boulter found again evidences of the existence of a cross-overexistence of a cross-over

G. Miller, C.J. Boulter, PRE 67, 2003

Cross over x associated with the probability of findingprobability of finding an avalanche an avalanche

with s > 1with s > 1 , lower bound for c (concentrate on 0.20 < 0.25)

• 0.12 x 0.16

• the result was not influenced by increasing the precision

• above this cross over , if > x < Fsc> >1 ( 10 -28)

However...However...

Their results have shown a qualitatively different behavior for qualitatively different behavior for the conservative regimethe 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 universal features in the non conserving regimeregime

Branching rate approachBranching 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)

c

Almost criticalO. Kinouchi, C.P.C. Prado, PRE 59 (1999)

J. X. de Carvalho, C. P. C. Prado,

Phys. Rev. Lett. 84 , 006, (2000).

One counts the number of supercritical descendents generated

when a site topples

Remains controversialRemains controversial alternative extrapolation procedures

Christensen et al, PRL 87 (2001)

de Carvalho and C.P.C. Prado, PRL 87 (2001)

Branching rate in OFC and R-OFCBranching rate in OFC and R-OFC

Layer branching rate Layer branching rate i i ( ( , L),, L),

i = 1, ... L/2 indicates the distance of the site from the boundary

• L = 1000 (non-conservative), L=700 (conservative)• c = 0.25• average avalanche sizes: (, L) = 1 - 1/s(, L). • “Control” models (beginning of organization)

Miller and Boulter, PRE 66, (2002)

The The qualitative difference in the behaviorqualitative difference in the behavior of the conservative of the conservative and non-conservative regimeand non-conservative regime was also observed in other was also observed in other

situationssituations

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!but only in the non-conservative regime!

Transients:Transients: J. X. de Carvalho, C.P.C. Prado, Physica A, 321 (2003)

Conservative caseConservative caseStationary state is identified by the mean value of the energy per site

After a transient, <Fi,j> fluctuates around a mean value

• The beginning of stationary state is clearly identified

• Transient time scales with L2

Conservative case

red line L = 100

black line L = 400

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

• Large fluctuations

• Much longer transient, scales Lb, b > 2 (in this case ~ 4)

• Initial “bump” scales L2

Non -conservative case, = 0.240

red line L = 100

black line L = 200

Non -conservative case, = 0.240

red line L = 100

black line L = 200

Detail of beginning...

Non -conservative case, = 0.249

red line L = 100

black line L = 400

Lattice must be large enough to show the “bump” ...

Dynamics of the epicentersDynamics 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 typeFree-scale behavior of Barabási-Albert type

Complex networks describe a wide range of systems in nature and society

R. Albert, A-L. Barabási, Rev. Mod. Phys. 74 (2002)

Free-scale networkFree-scale networkdegree of the node (connectivity)

P(k) ~ k -

Random graphRandom graph distribution is Poisson

Studied the OFC model in this contextStudied the OFC model in this context

Tiago P. Peixoto, C. P. C. Prado, 2003

0.249

L = 200, transients of 10 7, statistics of 10 5

0.240

Clear scalingClear scaling

Shifted upwards for the sake of clarity

Qualitative diference between conservative and Qualitative diference between conservative and non-conservative regimesnon-conservative regimes

0.249

0..25

L = 300

L = 200

We need a growing network ...We need a growing network ...

Different cell sizesDifferent cell sizes

L = 200,

1 X 1

L = 400,

2 X 2

Distribution of connectivity

L = 200, = 0.249

L = 200, = 0.25

ConclusionsConclusions

• Robustness of OFC model to describe real earthquakes

•Dynamics of epicenters

•Many open questions