2. Criticality in the Olami-Feder-Christensen model: Transients and Epicenters VIII Latin American Workshop on Nonlinear Phenomena LAWNP ’03 - Salvador, Bahia, 2003 Carmen P. C. Prado Universidade de São Paulo (prado@if.usp.br)

3. 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 )

4. Introduction SOC & Olami-Feder-Christensen model (OFC) History Recent developments Recent results Transients Epicenters (real earthquaques) Our results

5. 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 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

6. 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?

7. Earthquake dynamics is probably the best “experimental ” realization of SOC ideas... Exhibits universal power - laws Gutemberg-Richter ’s law Gutemberg-Richter ’s law (energy) P(E)  E -B Omori ’s law Omori ’s law (aftershocks and foreshocks) n(t) ~ t -A Two distinct time scales& punctuated equilibrium Two distinct time scales & punctuated equilibrium Slow: Slow: movement of tectonic plates (years) Fast: Fast: earthquakes (seconds) 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) )

8. 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...

9. Burridge-Knopoff model (1967) Burridge-Knopoff model (1967) Fixed plate Moving plate V  k i - 1 i i + 1 atri to Olami et al, PRL68 (92); Christensen et al, PRA 46 (92)

10. Perturbation: If some site becomes “active”, that is, if F > F th, the system relaxes: Relaxation: If any of the 4 neighbors exceeds F th, the relaxation rule is repeated. This process goes on until F < F th again for all sites of the lattice

11. The size distribution of avalanches obeys a power-law, reproducing the Gutemberg-Richter law Simulation for lattices of sizes L = 50,100 e 200. Conservative case:  = 1/4 SOC even in the non conservative regime

12. 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

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

14. First large scale simulation (Grassberger, PRE 49 (1994), Middleton & Tang, PRL 74 (1995) ) periodic boundary conditions: stationary state is periodic periodic boundary conditions: stationary state is periodic cross over (  ~ 0.18 ): small , ordered, period = L 2, dominated by small avalanches ( s=1) large , still periodic but disordered state 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;

15. Josué X. Carvalho Spatial correlation starts from the borders Random initial configuration

16. Josué X. Carvalho After 2 x 10 5 avalanches Spatial correlation starts from the borders

17. Josué X. Carvalho After 10 x 10 8 avalanches Spatial correlation starts from the borders

18. More recent work 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; infinite floating point precisionL   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.

19. 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). L   infinite precision In the limits L  , infinite precision, also all avalanches are of size1 (all sites topples with F = F th ).

20. In 2003, Miller and Boulter found again evidences of the existence of a cross-over G. Miller, C.J. Boulter, PRE 67, 2003 probability of finding an avalanche with s > 1 Cross over  x associated with the probability of finding an avalanche with 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 >1 ( 10 -28 )

21. However... qualitatively different behavior for the conservative regime 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) universal features in the non conserving regime They observed universal features in the non conserving regime

22. 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) cc Almost critical O. Kinouchi, C.P.C. Prado, PRE 59 (1999)

23. 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 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-OFC

24. Layer branching rate  i ( , 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)

25. The qualitative difference in the behavior of the conservative and non-conservative regimewas also observed in other situations 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!

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

27. Conservative case Stationary state is identified by the mean value of the energy per site After a transient, fluctuates around a mean value The beginning of stationary state is clearly identified Transient time scales with L 2 Conservative case red line L = 100 black line L = 400

28. conservative and non-conservative regimes display qualitatively different behavior during transient Large fluctuations Much longer transient, scales L b, b > 2 (in this case ~ 4) Initial “bump” scales L 2 Non -conservative case,  = 0.240 red line L = 100 black line L = 200

29. 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”...

30. 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

31. Complex networks describe a wide range of systems in nature and society R. Albert, A-L. Barabási, Rev. Mod. Phys. 74 (2002) Free-scale network degree of the node (connectivity) P(k) ~ k -  Random graph distribution is Poisson

32. Studied 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 scaling Shifted upwards for the sake of clarity

33. Qualitative diference between conservative and non-conservative regimes 0.249 0..25

34. L = 300 L = 200 We need a growing network...

35. Different cell sizes L = 200, 1 X 1 L = 400, 2 X 2

36. Distribution of connectivity L = 200,  = 0.249 L = 200,  = 0.25

37. Conclusions Robustness of OFC model to describe real earthquakes Dynamics of epicenters Many open questions