Abstract
We apply here the natural time analysis to the time series of the avalanches in several SOC models as well as to other dynamical models. First, in a simple deterministic SOC system introduced to describe avalanches in stick–slip phenomena that belongs to the same universality class as the “train” model for earthquakes introduced by Burridge and Knopoff, we find that the value K1 = 0.070 can be considered as quantifying the extent of the organization of the system at the onset of the critical stage. Second, in the conservative case of the Olami–Feder–Christensen (OFC) earthquake model, the value K1 = 0.070 is accompanied by an abrupt exponential increase of the avalanche size which is indicative of the approach to a critical behavior. In the non-conservative case of OFC, in the later part of the transient period, coherent domains of the strain field gradually develop accompanied by K1 values close to 0.070. Furthermore, there is a non-zero change ΔS of the entropy in natural time under time reversal, thus reflecting predictability in the OFC model. Third, an explanation for the validity of the condition K1 = 0.070 for critical systems on the basis of the dynamic scaling hypothesis is forwarded. Fourth, when quenching the 2D Ising model at temperatures close to but below Tc, which is qualitatively similar with the pressure stimulated currents SES generation model, and set Qk = |Mk| (where Mk stands for the evolution of the magnetization per spin), we find K1 = 0.070. Fifth, in a deterministic version of the original Bak–Tang–Wiesenfeld sandpile model, the value K1 ≈ 0.070 is reached during the transient to the self-organized criticality. Finally, natural time analysis of the avalanches observed in laboratory experiments on three-dimensional ricepiles and on the penetration of the magnetic flux into thin films of high Tc superconductors, leads to K1 values around K1 = 0.070. A further investigation of the experiment on ricepiles reveals that the sequential order of the avalanches captured by the natural time analysis is of profound importance for establishing the SOC state and constitutes the basis for the observation of the result K1 ≈ 0.070.
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References
Aegerter, C.M., Gunther, R., Wijngaarden, R.J.: Avalanche dynamics, surface roughening, and selforganized criticality: Experiments on a three-dimensional pile of rice. Phys. Rev. E 67, 051306 (2003)
Aegerter, C.M., Lorincz, K.A., Welling, M.S., Wijngaarden, R.J.: Extremal dynamics and the approach to the critical state: Experiments on a three dimensional pile of rice. Phys. Rev. Lett. 92, 058702 (2004)
Aegerter, C.M., Welling, M.S., Wijngaarden, R.J.: Self-organized criticality in the bean state in YBa2Cu3O7−x thin films. Europhys. Lett. 65, 753–759 (2004)
Altshuler, E., Johansen, T.H.: Colloquium: Experiments in vortex avalanches. Rev. Mod. Phys. 76, 471–487 (2004)
Bach, M., Wissel, F., Drossel, B.: Olami–Feder–Christensen model with quenched disorder. Phys. Rev. E 77(6), 067101 (2008)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)
Binney, J.J., Dowrick, N.J., Fisher, A.J., Newman, M.E.J.: The Theory of Critical Phenomena – An Introduction to the Renormalization Group. Oxford University Press, UK (1992)
Bonachela, J.A., M˜unoz, M.A.: Self-organization without conservation: true or just apparent scaleinvariance? J. Stat. Mech. P09009 (2009)
Boulter, C.J., Miller, G.: Nonuniversality and scaling breakdown in a nonconservative earthquake model. Phys. Rev. E 68, 056108 (2003)
Bray, A.J.: Theory of phase-ordering kinetics. Adv. Phys. 43, 357–459 (1994)
Bray, A.J., Briant, A.J., Jervis, D.K.: Breakdown of scaling in the nonequilibrium critical dynamics of the two-dimensional XY model. Phys. Rev. Lett. 84, 1503–1506 (2000)
Br¨oker, H.M., Grassberger, P.: Random neighbor theory of the Olami–Feder–Christensen earthquake model. Phys. Rev. E 56, 3944–3952 (1997)
Burridge, R., Knopoff, L.: Model and theoretical seismicity. Bull. Seismol. Soc. Am. 57, 341–371 (1967)
Campbell, A.M., Evetts, J.E.: Flux vortices and transport currents in type II superconductors. Adv Phys. 50, 1249–1449 (2001)
Carbone, A., Castelli, G., Stanley, H.E.: Analysis of clusters formed by the moving average of a long-range correlated time series. Phys. Rev. E 69, 026105 (2004)
Carbone, A., Stanley, H.E.: Directed self-organized critical patterns emerging from fractional Brownian paths. Physica A 340, 544–551 (2004)
Carlson, J.M., Langer, J.S.: Properties of earthquakes generated by fault dynamics. Phys. Rev. Lett. 62, 2632–2635 (1989)
Carlson, J.M., Langer, J.S., Shaw, B.E.: Dynamics of earthquake faults. Rev. Mod. Phys. 66, 657– 670 (1994)
Caruso, F., Pluchino, A., Latora, V., Vinciguerra, S., Rapisarda, A.: Analysis of self-organized criticality in the Olami–Feder–Christensen model and in real earthquakes. Phys. Rev. E 75, 055101 (2007)
de Carvalho, J.X., Prado, C.P.C.: Self-organized criticality in the Olami–Feder–Christensen model. Phys. Rev. Lett. 84, 4006–4009 (2000)
de Carvalho, J.X., Prado, C.P.C.: Dealing with transients in models with self-organized criticality. Physica A 321, 519–528 (2003)
Ceva, H.: Influence of defects in a coupled map lattice modeling earthquakes. Phys. Rev. E 52, 154–158 (1995)
Chabanol, M.L., Hakim, V.: Analysis of a dissipative model of self-organized criticality with random neighbors. Phys. Rev. E 56, R2343–R2346 (1997)
Chen, M.J., Lu, M.P.: On-off switching of edge direct tunneling currents in metal-oxidesemiconductor field-effect transistors. Appl. Phys. Lett.. 81, 3488–3490 (2002)
Davidsen, J., Paczuski, M.: 1/ f α noise from correlations between avalanches in self-organized criticality. Phys. Rev. E 66, 050101 (2002)
Dhar, D.: Theoretical studies of self-organized criticality. Physica A 369, 29–70 (2006)
Dhar, D., Ramaswamy, R.: Exactly solved model of self-organized critical phenomena. Phys. Rev. Lett. 63, 1659–1662 (1989)
Drossel, B.: Complex scaling behavior of nonconserved self-organized critical systems. Phys. Rev. Lett. 89, 238701 (2002)
Fawcett, T.: An introduction to ROC analysis. Pattern Recogn. Lett. 27, 861–874 (2006)
Fey, A., Levine, L., Peres, Y.: Growth rates and explosions in sandpiles. J. Stat. Phys. 138, 143–159 (2010)
Garber, A., Hallerberg, S., Kantz, H.: Predicting extreme avalanches in self-organized critical sandpiles. Phys. Rev. E 80, 026124 (2009)
Garber, A., Kantz, H.: Finite-size effects on the statistics of extreme events in the BTW model. Eur. Phys. J. B 67, 437–443 (2009)
de Gennes, P.G.: Superconductivity of Metals and Alloys. Addison-Wesley, New York (1966)
Grassberger, P.: Efficient large-scale simulations of a uniformly driven system. Phys. Rev. E 49, 2436–2444 (1994)
Helmstetter, A., Hergarten, S., Sornette, D.: Properties of foreshocks and aftershocks of the nonconservative self-organized critical Olami–Feder–Christensen model. Phys. Rev. E 70, 046120 (2004)
Helmstetter, A., Sornette, D.: Foreshocks explained by cascades of triggered seismicity. J. Geophys. Res. 108(B10), 2457 (2003)
Helmstetter, A., Sornette, D., Grasso, J.R.: Mainshocks are aftershocks of conditional foreshocks: How do foreshock statistical properties emerge from aftershock laws. J. Geophys. Res. 108(B1), 2046 (2003)
Hergarten, S., Neugebauer, H.J.: Foreshocks and aftershocks in the Olami–Feder–Christensen model. Phys. Rev. Lett. 88, 238501 (2002)
Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435– 479 (1977)
Huang, Y., Saleur, H., Sammis, C., Sornette, D.: Precursors, aftershocks, criticality and selforganized criticality. Europhys. Lett. 41, 43–48 (1998)
Ito, N.: Non-equilibrium relaxation and interface energy of the Ising model. Physica A 196, 591–614 (1993)
J´anosia, I.M., Kert´esz, J.: Self-organized criticality with and without conservation. Physica A 200, 179–188 (1993)
Jensen, H.J.: Self-organized criticality: emergent complex behavior in physical and biological systems. Cambridge University Press, New York (1998)
Keilis-Borok, V.I., Kossobokov, V.G.: Premonitory activation of earthquake flow: algorithm M8. Phys. Earth Planet. Inter. 61, 73–83 (1990)
Keilis-Borok, V.I., Rotwain, I.M.: Diagnosis of time of increased probability of strong earthquakes in different regions of the world: algorithm CN. Phys. Earth Planet. Inter. 61, 57–72 (1990)
Ktitarev, D.V., L¨ubeck, S., Grassberger, P., B. Priezzhev, V.: Scaling of waves in the Bak–Tang– Wiesenfeld sandpile model. Phys. Rev. E 61, 81–92 (2000)
Landau, D.P., Binder, K.: A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge Univ. Press, New York (2005)
Lee, M.J.: Pseudo-random-number generators and the square site percolation threshold. Phys. Rev. E 78, 031131 (2008)
Lippiello, E., de Arcangelis, L., Godano, C.: Influence of time and space correlations on earthquake magnitude. Phys. Rev. Lett. 100, 038501 (2008)
Lippiello, E., Godano, C., de Arcangelis, L.: Dynamical scaling in branching models for seismicity. Phys. Rev. Lett. 98, 098501 (2007)
Lise, S., Paczuski, M.: Self-organized criticality and universality in a nonconservative earthquake model. Phys. Rev. E 63, 036111 (2001)
Middleton, A.A., Tang, C.: Self-organized criticality in nonconserved systems. Phys. Rev. Lett. 74, 742–745 (1995)
Miller, G., Boulter, C.J.: Measurements of criticality in the Olami–Feder–Christensen model. Phys. Rev. E 66, 016123 (2002)
Mori, T., Kawamura, H.: Simulation study of earthquakes based on the two-dimensional Burridge- Knopoff model with long-range interactions. Phys. Rev. E 77, 051123 (2008)
Mousseau, N.: Synchronization by disorder in coupled systems. Phys. Rev. Lett. 77, 968–971 (1996)
Nakanishi, H.: Statistical properties of the cellular-automaton model for earthquakes. Phys. Rev. A 43, 6613–6621 (1991)
O´ dor, G.: Universality classes in nonequilibrium lattice systems. Rev. Mod. Phys. 76, 663–724 (2004)
Olami, Z., Feder, H.J.S., Christensen, K.: Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett. 68, 1244–1247 (1992)
Paczuski, M., Maslov, S., Bak, P.: Avalanche dynamics in evolution, growth, and depinning models. Phys. Rev. E 53, 414–443 (1996)
Peixoto, T.P., Davidsen, J.: Network of recurrent events for the Olami–Feder–Christensen model. Phys. Rev. E 77, 066107 (2008)
Pepke, S.L., Carlson, J.M.: Predictability of self-organizing systems. Phys. Rev. E 50, 236–242 (1994)
Pepke, S.L., Carlson, J.M., Shaw, B.E.: Prediction of large events on a dynamical model of a fault. J. Geophys. Res. 99(B4), 6769–6788 (1994)
P´erez, C.J., Corral, A., D´ıaz-Guilera, A., Christensen, K., Arenas, A.: On self-organized criticality and synchronization in lattice models of coupled dynamical systems. Int. J. Mod. Phys. B 10, 1111– 1151 (1996)
Ramos, O., Altshuler, E.,M˚aløy, K.J.: Quasiperiodic events in an earthquake model. Phys. Rev. Lett. 96, 098501 (2006)
Ramos, O., Altshuler, E., M˚aløy, K.J.: Avalanche prediction in a self-organized pile of beads. Phys. Rev. Lett. 102, 078701 (2009)
Sammis, C.G., Smith, S.W.: Seismic cycles and the evolution of stress correlation in cellular automaton models of finite fault networks. Pure Appl. Geophys. 155, 307–334 (1999)
Sarlis, N.V., Skordas, E.S., Varotsos, P.A.: Natural time analysis of the Olami–Feder–Christensen model. (to be published) (2011)
Sarlis, N.V., Skordas, E.S., Varotsos, P.A.: Nonextensivity and natural time: The case of seismicity. Phys. Rev. E 82, 021110 (2010)
Sarlis, N.V., Varotsos, P.A., Skordas, E.S.: Flux avalanches in YBa2Cu3O7−x films and rice piles: Natural time domain analysis. Phys. Rev. B 73, 054504 (2006)
Scholz, C.H.: Earthquakes and friction laws. Nature (London) 391, 37–42 (1998)
Scholz, C.H.: The Mechanics of Earthquakes and Faulting, 2nd ed. Cambridge University Press, Cambridge U.K. (2002)
Sicilia, A., Arenzon, J.J., Bray, A.J., Cugliandolo, L.F.: Domain growth morphology in curvaturedriven two-dimensional coarsening. Phys. Rev. E 76, 061116 (2007)
Sornette, D.: Critical Phenomena in Natural Science, 2nd edn. Springer, Berlin (2004)
de Sousa Vieira, M.: Self-organized criticality in a deterministic mechanical model. Phys. Rev. A 46, 6288–6293 (1992)
de Sousa Vieira, M.: Simple deterministic self-organized critical system. Phys. Rev. E 61, R6056– R6059 (2000)
Su, H., Welch, D.O., Wong-Ng, W.: Strain effects on point defects and chain-oxygen order-disorder transition in 123 cuprate compounds. Phys. Rev. B 70, 054517 (2004)
Varotsos, P., Alexopoulos, K.: Thermodynamics of Point Defects and their Relation with Bulk Properties. North Holland, Amsterdam (1986)
Varotsos, P., Ludwig, W., Alexopoulos, K.: Calculation of the formation volume of vacancies in solids. Phys. Rev. B 18, 2683–2691 (1978)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Lazaridou, M.S.: Entropy in natural time domain. Phys. Rev. E 70, 011106 (2004)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Uyeda, S., Kamogawa, M.: Natural time analysis of critical phenomena. under preparation (2011)
Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Uyeda, S., Kamogawa, M.: Natural time analysis of critical phenomena. the case of seismicity. EPL 92, 29002 (2010)
Welling, M.S., Aegerter, C.M., Wijngaarden, R.J.: Self-organized criticality induced by quenched disorder: Experiments on flux avalanches in NbHx films. Phys. Rev. B 71, 104515 (2005)
Wiesenfeld, K., Theiler, J., McNamara, B.: Self-organized criticality in a deterministic automaton. Phys. Rev. Lett. 65, 949–952 (1990)
Wissel, F., Drossel, B.: Transient and stationary behavior of the Olami–Feder–Christensen model. Phys. Rev. E 74, 066109 (2006)
Yang, X., Du, S., Ma, J.: Do earthquakes exhibit self-organized criticality? Phys. Rev. Lett. 92, 228501 (2004)
Zaitsev, S.I.: Robin Hood as self-organized criticality. Physica A 189, 411–416 (1992)
Zhang, S., Huang, Z., Ding, E.: Predictions of large events on a spring-block model. J. Phys. A: Math. Gen. 29, 4445–4455 (1996)
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Varotsos, P.A., Sarlis, N.V., Skordas, E.S. (2011). Natural Time Analysis of Dynamical Models. In: Natural Time Analysis: The New View of Time. Springer Praxis Books(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16449-1_8
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