Dynamics of epicenters in the Dynamics of epicenters in the

Olami-Feder-Christensen ModelOlami-Feder-Christensen Model

Trends and Perspectives in Non-extensive Statistical MechanicsTrends and Perspectives in Non-extensive Statistical Mechanics6060thth-birthday of C. Tsallis-birthday of C. Tsallis

Angra dos Reis, Rio de Janeiro, 2003Angra dos Reis, Rio de Janeiro, 2003

Carmen P. C. Prado Universidade de São Paulo

(prado@if.usp.br)

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))

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

Introduction:Introduction:• Earthquakes, SOC and the Olami-Feder-Earthquakes, SOC and the Olami-Feder-Christensen model (OFC)Christensen model (OFC)

Recent results on earthquake dynamics:Recent results on earthquake dynamics:• Epicenter distribution (real earthquakes)Epicenter distribution (real earthquakes)

• Epicenters in the OFC model (our results)Epicenters in the OFC model (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

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 friction

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

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

Relaxation:Relaxation: 0),( jiF

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

Perturbation:Perturbation: ),(),( jiFjiF

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

(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

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 and Omori’s Lawand Omori’s Law

Simulation for lattices of sizes L = 50,100 e 200.

Conservative case: = 1/4

SOC even in the non SOC even in the non conservative regimeconservative regime

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 but only in the non-conservative regime!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

= 0 model is non-critical = 0.25 model is criticalat which value of = c the system changes its behavior ???

• 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 ?

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 ().

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).

Almost criticalRemains controversialRemains controversial

Dynamics of the epicentersDynamics of the epicenters

SS. Abe, N. Suzuki, cond-matt / 0210289. Abe, N. Suzuki, cond-matt / 0210289

• Instead of the spatial distribution (that Instead of the spatial distribution (that is fractal) , the looked at the time is fractal) , the looked at the time evolution of epicenters evolution of epicenters

•Found a new scaling law for Found a new scaling law for earthquakes (Japan and South earthquakes (Japan and South California)California)

Fractal distribution

SS. Abe, N. Suzuki, . Abe, N. Suzuki, cond-matt / 0210289cond-matt / 0210289

TimeTime sequence sequence of epicenters from of epicenters from earthquake data of a district of earthquake data of a district of southern California and Japansouthern California and Japan

• area was divided into small cubic small cubic cellscells, each of which is regarded as vertex of a graph if an epicenter occurs in it;

• the seismic data was mapped into the seismic data was mapped into na evolving random graph;na evolving random graph;

Free-scale behavior of Barabási-Albert typeFree-scale behavior of Barabási-Albert type

S. Abe, N. Suzuki, cond-matt / 0210289S. Abe, N. Suzuki, cond-matt / 0210289

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

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

Free-scale networkFree-scale networkconnectivity of the node

P(k) ~ k -

Random graphRandom graph distribution is Poisson

We studied the OFC model in this context, to see if it We studied the OFC model in this context, to see if it was able to predict also this behaviorwas able to predict also this behavior

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

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

Clear scalingClear scaling

( Curves were shifted upwards for the sake of clarity )

0.240

0.249

The exponent The exponent that characterizes the power-law that characterizes the power-law behavior of P(k), for different values of behavior of P(k), for different values of

L = 200,

1 X 1

L = 400,

2 X 2

The size of the cell does not affect the connectivity The size of the cell does not affect the connectivity distribution P(k) ... distribution P(k) ...

But surprisingly, But surprisingly, There is a qualitative diference between There is a qualitative diference between conservative and non-conservative regimes !conservative and non-conservative regimes !

0..25

L = 300

L = 200

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

Distribution of connectivity

L = 200, = 0.249

L = 200, = 0.25

Spatial distribution of Spatial distribution of connectivity, (connectivity, (non-conservative)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 effectIt is not a boundary effect

Spatial distribution of Spatial distribution of connectivity, (connectivity, (conservative)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 homogeneousMuch more homogeneous

ConclusionsConclusions

• Robustness of OFC model to describe real earthquakes, since its Robustness of OFC model to describe real earthquakes, since its able to reproduce the scale free network observed in real dataable to reproduce the scale free network observed in real data

• New dynamical mechanism to generate a free-scale network, The New dynamical mechanism to generate a free-scale network, The preferential attachment present in the network is not a rule but a preferential attachment present in the network is not a rule but a signature of the dynamicssignature of the dynamics

• Indicates (in agreement with many previous works) qualitatively Indicates (in agreement with many previous works) qualitatively different behavior between conservative and non-conservative different behavior between conservative and non-conservative modelsmodels

• Many open questions...Many open questions...