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Natural Time Analysis: The New View of Time pp 341–380Cite as

Natural Time Analysis of Dynamical Models

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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

  1. 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)

    Article  Google Scholar 

  2. 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)

    Article  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. Altshuler, E., Johansen, T.H.: Colloquium: Experiments in vortex avalanches. Rev. Mod. Phys. 76, 471–487 (2004)

    Article  Google Scholar 

  5. Bach, M., Wissel, F., Drossel, B.: Olami–Feder–Christensen model with quenched disorder. Phys. Rev. E 77(6), 067101 (2008)

    Article  Google Scholar 

  6. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)

    Article  Google Scholar 

  7. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)

    Google Scholar 

  8. 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)

    Google Scholar 

  9. Bonachela, J.A., M˜unoz, M.A.: Self-organization without conservation: true or just apparent scaleinvariance? J. Stat. Mech. P09009 (2009)

    Google Scholar 

  10. Boulter, C.J., Miller, G.: Nonuniversality and scaling breakdown in a nonconservative earthquake model. Phys. Rev. E 68, 056108 (2003)

    Article  Google Scholar 

  11. Bray, A.J.: Theory of phase-ordering kinetics. Adv. Phys. 43, 357–459 (1994)

    Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. Br¨oker, H.M., Grassberger, P.: Random neighbor theory of the Olami–Feder–Christensen earthquake model. Phys. Rev. E 56, 3944–3952 (1997)

    Google Scholar 

  14. Burridge, R., Knopoff, L.: Model and theoretical seismicity. Bull. Seismol. Soc. Am. 57, 341–371 (1967)

    Google Scholar 

  15. Campbell, A.M., Evetts, J.E.: Flux vortices and transport currents in type II superconductors. Adv Phys. 50, 1249–1449 (2001)

    Article  Google Scholar 

  16. 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)

    Article  Google Scholar 

  17. Carbone, A., Stanley, H.E.: Directed self-organized critical patterns emerging from fractional Brownian paths. Physica A 340, 544–551 (2004)

    Article  Google Scholar 

  18. Carlson, J.M., Langer, J.S.: Properties of earthquakes generated by fault dynamics. Phys. Rev. Lett. 62, 2632–2635 (1989)

    Article  Google Scholar 

  19. Carlson, J.M., Langer, J.S., Shaw, B.E.: Dynamics of earthquake faults. Rev. Mod. Phys. 66, 657– 670 (1994)

    Article  Google Scholar 

  20. 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)

    Article  Google Scholar 

  21. de Carvalho, J.X., Prado, C.P.C.: Self-organized criticality in the Olami–Feder–Christensen model. Phys. Rev. Lett. 84, 4006–4009 (2000)

    Article  Google Scholar 

  22. de Carvalho, J.X., Prado, C.P.C.: Dealing with transients in models with self-organized criticality. Physica A 321, 519–528 (2003)

    Article  Google Scholar 

  23. Ceva, H.: Influence of defects in a coupled map lattice modeling earthquakes. Phys. Rev. E 52, 154–158 (1995)

    Article  Google Scholar 

  24. Chabanol, M.L., Hakim, V.: Analysis of a dissipative model of self-organized criticality with random neighbors. Phys. Rev. E 56, R2343–R2346 (1997)

    Article  Google Scholar 

  25. 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)

    Article  Google Scholar 

  26. Davidsen, J., Paczuski, M.: 1/ f α noise from correlations between avalanches in self-organized criticality. Phys. Rev. E 66, 050101 (2002)

    Article  Google Scholar 

  27. Dhar, D.: Theoretical studies of self-organized criticality. Physica A 369, 29–70 (2006)

    Article  Google Scholar 

  28. Dhar, D., Ramaswamy, R.: Exactly solved model of self-organized critical phenomena. Phys. Rev. Lett. 63, 1659–1662 (1989)

    Article  Google Scholar 

  29. Drossel, B.: Complex scaling behavior of nonconserved self-organized critical systems. Phys. Rev. Lett. 89, 238701 (2002)

    Article  Google Scholar 

  30. Fawcett, T.: An introduction to ROC analysis. Pattern Recogn. Lett. 27, 861–874 (2006)

    Article  Google Scholar 

  31. Fey, A., Levine, L., Peres, Y.: Growth rates and explosions in sandpiles. J. Stat. Phys. 138, 143–159 (2010)

    Article  Google Scholar 

  32. Garber, A., Hallerberg, S., Kantz, H.: Predicting extreme avalanches in self-organized critical sandpiles. Phys. Rev. E 80, 026124 (2009)

    Article  Google Scholar 

  33. 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)

    Article  Google Scholar 

  34. de Gennes, P.G.: Superconductivity of Metals and Alloys. Addison-Wesley, New York (1966)

    Google Scholar 

  35. Grassberger, P.: Efficient large-scale simulations of a uniformly driven system. Phys. Rev. E 49, 2436–2444 (1994)

    Article  Google Scholar 

  36. 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)

    Article  Google Scholar 

  37. Helmstetter, A., Sornette, D.: Foreshocks explained by cascades of triggered seismicity. J. Geophys. Res. 108(B10), 2457 (2003)

    Google Scholar 

  38. 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)

    Google Scholar 

  39. Hergarten, S., Neugebauer, H.J.: Foreshocks and aftershocks in the Olami–Feder–Christensen model. Phys. Rev. Lett. 88, 238501 (2002)

    Article  Google Scholar 

  40. Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435– 479 (1977)

    Article  Google Scholar 

  41. Huang, Y., Saleur, H., Sammis, C., Sornette, D.: Precursors, aftershocks, criticality and selforganized criticality. Europhys. Lett. 41, 43–48 (1998)

    Article  Google Scholar 

  42. Ito, N.: Non-equilibrium relaxation and interface energy of the Ising model. Physica A 196, 591–614 (1993)

    Article  Google Scholar 

  43. J´anosia, I.M., Kert´esz, J.: Self-organized criticality with and without conservation. Physica A 200, 179–188 (1993)

    Google Scholar 

  44. Jensen, H.J.: Self-organized criticality: emergent complex behavior in physical and biological systems. Cambridge University Press, New York (1998)

    Google Scholar 

  45. Keilis-Borok, V.I., Kossobokov, V.G.: Premonitory activation of earthquake flow: algorithm M8. Phys. Earth Planet. Inter. 61, 73–83 (1990)

    Article  Google Scholar 

  46. 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)

    Article  Google Scholar 

  47. 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)

    Google Scholar 

  48. Landau, D.P., Binder, K.: A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge Univ. Press, New York (2005)

    Book  Google Scholar 

  49. Lee, M.J.: Pseudo-random-number generators and the square site percolation threshold. Phys. Rev. E 78, 031131 (2008)

    Article  Google Scholar 

  50. Lippiello, E., de Arcangelis, L., Godano, C.: Influence of time and space correlations on earthquake magnitude. Phys. Rev. Lett. 100, 038501 (2008)

    Article  Google Scholar 

  51. Lippiello, E., Godano, C., de Arcangelis, L.: Dynamical scaling in branching models for seismicity. Phys. Rev. Lett. 98, 098501 (2007)

    Article  Google Scholar 

  52. Lise, S., Paczuski, M.: Self-organized criticality and universality in a nonconservative earthquake model. Phys. Rev. E 63, 036111 (2001)

    Article  Google Scholar 

  53. Middleton, A.A., Tang, C.: Self-organized criticality in nonconserved systems. Phys. Rev. Lett. 74, 742–745 (1995)

    Article  Google Scholar 

  54. Miller, G., Boulter, C.J.: Measurements of criticality in the Olami–Feder–Christensen model. Phys. Rev. E 66, 016123 (2002)

    Article  Google Scholar 

  55. 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)

    Article  Google Scholar 

  56. Mousseau, N.: Synchronization by disorder in coupled systems. Phys. Rev. Lett. 77, 968–971 (1996)

    Article  Google Scholar 

  57. Nakanishi, H.: Statistical properties of the cellular-automaton model for earthquakes. Phys. Rev. A 43, 6613–6621 (1991)

    Article  Google Scholar 

  58. O´ dor, G.: Universality classes in nonequilibrium lattice systems. Rev. Mod. Phys. 76, 663–724 (2004)

    Google Scholar 

  59. 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)

    Article  Google Scholar 

  60. Paczuski, M., Maslov, S., Bak, P.: Avalanche dynamics in evolution, growth, and depinning models. Phys. Rev. E 53, 414–443 (1996)

    Article  Google Scholar 

  61. Peixoto, T.P., Davidsen, J.: Network of recurrent events for the Olami–Feder–Christensen model. Phys. Rev. E 77, 066107 (2008)

    Article  Google Scholar 

  62. Pepke, S.L., Carlson, J.M.: Predictability of self-organizing systems. Phys. Rev. E 50, 236–242 (1994)

    Article  Google Scholar 

  63. 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)

    Google Scholar 

  64. 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)

    Google Scholar 

  65. Ramos, O., Altshuler, E.,M˚aløy, K.J.: Quasiperiodic events in an earthquake model. Phys. Rev. Lett. 96, 098501 (2006)

    Google Scholar 

  66. Ramos, O., Altshuler, E., M˚aløy, K.J.: Avalanche prediction in a self-organized pile of beads. Phys. Rev. Lett. 102, 078701 (2009)

    Google Scholar 

  67. 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)

    Article  Google Scholar 

  68. Sarlis, N.V., Skordas, E.S., Varotsos, P.A.: Natural time analysis of the Olami–Feder–Christensen model. (to be published) (2011)

    Google Scholar 

  69. Sarlis, N.V., Skordas, E.S., Varotsos, P.A.: Nonextensivity and natural time: The case of seismicity. Phys. Rev. E 82, 021110 (2010)

    Article  Google Scholar 

  70. 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)

    Article  Google Scholar 

  71. Scholz, C.H.: Earthquakes and friction laws. Nature (London) 391, 37–42 (1998)

    Google Scholar 

  72. Scholz, C.H.: The Mechanics of Earthquakes and Faulting, 2nd ed. Cambridge University Press, Cambridge U.K. (2002)

    Google Scholar 

  73. 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)

    Article  Google Scholar 

  74. Sornette, D.: Critical Phenomena in Natural Science, 2nd edn. Springer, Berlin (2004)

    Google Scholar 

  75. de Sousa Vieira, M.: Self-organized criticality in a deterministic mechanical model. Phys. Rev. A 46, 6288–6293 (1992)

    Google Scholar 

  76. de Sousa Vieira, M.: Simple deterministic self-organized critical system. Phys. Rev. E 61, R6056– R6059 (2000)

    Google Scholar 

  77. 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)

    Article  Google Scholar 

  78. Varotsos, P., Alexopoulos, K.: Thermodynamics of Point Defects and their Relation with Bulk Properties. North Holland, Amsterdam (1986)

    Google Scholar 

  79. Varotsos, P., Ludwig, W., Alexopoulos, K.: Calculation of the formation volume of vacancies in solids. Phys. Rev. B 18, 2683–2691 (1978)

    Article  Google Scholar 

  80. Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Lazaridou, M.S.: Entropy in natural time domain. Phys. Rev. E 70, 011106 (2004)

    Article  Google Scholar 

  81. Varotsos, P.A., Sarlis, N.V., Skordas, E.S., Uyeda, S., Kamogawa, M.: Natural time analysis of critical phenomena. under preparation (2011)

    Google Scholar 

  82. 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)

    Google Scholar 

  83. 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)

    Article  Google Scholar 

  84. Wiesenfeld, K., Theiler, J., McNamara, B.: Self-organized criticality in a deterministic automaton. Phys. Rev. Lett. 65, 949–952 (1990)

    Article  Google Scholar 

  85. Wissel, F., Drossel, B.: Transient and stationary behavior of the Olami–Feder–Christensen model. Phys. Rev. E 74, 066109 (2006)

    Article  Google Scholar 

  86. Yang, X., Du, S., Ma, J.: Do earthquakes exhibit self-organized criticality? Phys. Rev. Lett. 92, 228501 (2004)

    Article  Google Scholar 

  87. Zaitsev, S.I.: Robin Hood as self-organized criticality. Physica A 189, 411–416 (1992)

    Article  Google Scholar 

  88. Zhang, S., Huang, Z., Ding, E.: Predictions of large events on a spring-block model. J. Phys. A: Math. Gen. 29, 4445–4455 (1996)

    Article  Google Scholar 

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  1. University of Athens, Athens, Greece

    Professor Panayiotis A. Varotsos, Dr. Nicholas V. Sarlis & Dr. Efthimios S. Skordas

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  1. Professor Panayiotis A. Varotsos
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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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