Complexity in Geophysics. A description based on Non-extensive Statistical Physics .

1. Laboratory of Geophysics & Seismology • Technological Educational Institute of Crete Complexity in Geophysics.A description based on Non-extensive Statistical Physics. Approaching a Complex World . • Filippos Vallianatos • UNESCO Chair on Solid Earth Physics and Geohazards Risk Reduction, • Technological Educational Institute of Crete • July 2017

2. Complexity and Earth(quake) Physics A description of Seismicity based on Non-extensive Statistical Physics: An introduction to Non-extensive Statistical Seismology.  Material of the presentation is based on works by Tsallis, Sotologo, Silva, Telesca, Vallianatos and co-workers

3. Complexity-Statistical Physics Multiscale Dynamics Earthquake Fault Systems

4. Systems composed of large number of simple, interacting elements Statistical Physics ………………………………… Multiscale Dynamics … Earthquake Fault Systems • Uninterested in small-scale (random) behaviour • Use methods of statistics (averages!) • Huge range of scale • Phenomenology of dynamics

5. Introduction and historical Background • Experimental evidence of a non-extensive statistical physics behaviour of fracture in triaxially deformed Etna basalt using acoustic emissions. • Non Extensive statistical physics approach to fault population distribution. • Seismicity in terms of non-extensive statistical physics. A challenge • Evidence of non-extensive thermodynamic lithospheric instability at the approach of the 2004 Sumatran- Andaman and 2011 Honshu mega-earthquakes • Is plate tectonics a case of non-extensive thermodynamics?What’saboutgeomagneticrevarsals ? • Is Earth’s ambient noise q-Gaussian distributed ? • Applications on Structures’ vibrations

6. Hierarchy - Scales A “system of systems”

7. Space-Time Process Scales

9. Concerning the physics of fracture, many questions have not yet been answered since the phenomenon is subjected to many uncertainties and degrees of freedom. Regarding the physics of “many” fractures [AE / earthquakes] and how this can be derived from first principles, one may wonder: • How can the collective properties of a set formed by all fractures/earthquakes in a given sample/region, be derived? • How does the patern formed by all fractures/ earthquakes, depend on its elementary constituents ? What are these properties?

10. What are the collective properties of the seismicity/fractures set, defined by all earthquakes [fractures] in a given region? • How does seismicity [in rocks AE or PSC], which is the “structure” formed by all earthquakes [fracturing], depend on its elementary constituents, the earthquakes [fractures]? Which are the properties of this dependence? • What kind of dynamic process does seismicity [AE and/or PSC] constitute? • How are laboratory observations of fracturing and creep related with geodynamic deformation and what are the accompanying phenomena?

11. It is natural then to consider that the physics of many fractures/earthquakes has to be studied with a different approach than the physics of one and in this sense we can consider the use of statistical physics not only appropriate but also necessary to understand the collective properties of earthquakes

12. Then a natural question arises. • What type of statistical physics is appropriate to commonly describe effects from the microscale and crack opening level to the level of large earthquakes and plate tectonics?

13. An answer to the previous question could be non-extensive statistical physics (NESP), originally introduced by Tsallis (1988). • The latter is strongly supported by the fact that this type of statistical mechanics is the appropriate methodological tool to describe entities with (multi) fractal distributions of their elements and where long-range interactions or intermittency are important, as in fracturing phenomena and earthquakes. • NESPis based on a generalization of the classic Boltzmann-Gibbs entropy and has the main advantage that it considers all-length scale correlations among the elements of a system, leading to an asymptotic power-law behavior. So far, NESP has found many applications in nonlinear dynamical systems including earthquakes. • In a series of recent publications, it has been shown that the collective properties of the earthquake and fault populations from the laboratory scale, to local, regional and global scale can well reproduced by means of NESP.

14. Nonextensive StatisticalPhysics • Physics is based on two pillars: • energy and entropy. • ENERGY concerns (dynamical or mechanical) possibilities; • ENTROPY concerns the probabilities of those possibilities. • Energy ….is more basic, and clearly depends on the physical system (classical, quantum, relativistic, or any other); • Entropy ……is more subtle, and reflects the information upon the physical system.

15. Statistical Mechanics • Entropy as a physical property was introduced by the German physicist Rudolf Clausius in the mid-1860s to explain the maximum energy available for useful work in heat engines. • It was not until the work of Boltzmann in the late 1870s, however, that entropy become clearly defined according to the famous formula: • , • where, S is entropy, kΒis the Boltzmann constant and W is the number of microstates available to a system. • Ludwig Boltzmann • (1844-1906) • Boltzmann’s formula shows that entropy increases logarithmically with the number of microstates. • It also tends to class entropy as an extensive property, that is a property, like volume or mass, whose value is proportional to the amount of matter in the system.

16. Ludwig Eduard Boltzmann, Austrian Physicist, (1844-1906) Josiah Willard Gibbs (1839 – 1903) American theoretical Physicisttician Constantino Tsallis (1943– ) is a Physicist working in Brazil. He was born in GREECE and grew up in Argentina

17. Tsallis generalized thermostatistics In the last decade a lot of effort has been devoted to understand if thermodynamics can be generalized to nonequilibrium complex systems In particular one of these attempts is that one started by Constantino Tsallis with his seminal paperJ. Stat. Phys. 52 (1988) 479 For recent reviews see for example: • C. Tsallis, Nonextensive Statistical Mechanics and Thermodynamics , Lecture Notes in Physics, eds. S. Abe and Y. Okamoto, Springer, Berlin, (2001); • Proceedings of NEXT2001, special issue of Physica A 305 (2002) eds. G. Kaniadakis, M. Lissia and A Rapisarda; • C. Tsallis, A. Rapisarda, V. Latora and F. Baldovin in "Dynamics and Thermodynamics of Systems with Long-Range Interactions", T. Dauxois, S. Ruffo,E. Arimondo, M. Wilkens eds., Lecture Notes in Physics Vol. 602, Springer (2002) 140; • ``Nonextensive Entropy - Interdisciplinary Applications'', C. Tsallis and M. Gell-Mann eds., Oxford University Press (2003). • A. Cho, Science 297 (2002) 1268; S. Abe and A.K. Rajagopal, Science 300, (2003)249; A. Plastino, Science 300 (2003) 250; V. Latora, A. Rapisarda and A. Robledo, Science 300 (2003) 250.

18. Although BG entropy seems the correct one to be used in a large and important class of physical systems with strongly chaotic dynamics (positive maximal Lyapunov exponent), an important class of weakly chaotic systems (where the maximal Lyapunov exponent vanishes) violates this hypothesis. • Additionally, if the effective microscopic interactions and memory are short-ranged (for instance Markovian processes) and the boundary conditions are smooth, then BG statistical mechanics seems to correctly describe nature. • On the other hand, if some or all of these restrictions are violated (long-range interactions, non-markovian microscopic memory, multifractal boundary conditions and multifractal structures), then another type of statistical mechanics seems appropriate to describe nature (see for instance Zaslavsky, 1999; Tsallis, 2001).

19. Tsallis entropy effective as a measure of nonextensive entropy successful in describing systems with long-range interactions, multifractal space–time constraints or long-term memory effects

20. Foundations and applications • Beck, Lewis and Swinney PRE 63 (2001) 035303R ; Beck, Physica A 295 (2001) 195 and cond-mat/0303288 Physica D (2003) in press. • F. Vallianatos, NHESS, 2009 Natural Hazards Turbulence High energy collisions • Wilk et al PRL 84 (2000) 2770, Beck Physica A 286 (2000) 164; Bediaga, Curado, De Miranda , Physica A 286 (2000) 156 Cosmic rays • C. Tsallis, J.C. Anjos and E.P. Borges, Phys. Lett. A 310 (2003) 372. Econophysics • L. Borland, Phys. Rev. Lett. 89 (2002) 098701 Biological systems • Upaddhyaya et al Physica A 293 (2001) 549 • V.Latora, M.Baranger, A. Rapisarda and C.Tsallis, Phys. Lett. A 273 (2000) 97; U. Tirnakli, Phys. Rev. E 66;(2002) 066212; E.P. Borges, C. Tsallis,G.F.J. Ananos, P.M.C. de Oliveira, Phys. Rev. Lett. 25 (2002) 254103. Maps at the edge of chaos For the most recent studies on the theoretical foundations see C. Beck, Phys Rev. Lett. 87 (2001) 180601 C. Beck and E.G.D. Cohen, Superstatistics Physica A 322 (2003) 267 F. Baldovin and A. Robledo, Phys. Rev. E 66 (2002) 045104(R) and Europhys. Lett. 60 (2002) 518 For recent successful applications of the generalized statistics see for example

21. Nonextensive Statisticalmechanics&fracture physics.…from cracks to faults…..and even more….

22. Boltzmann- Gibbs entropy • The Boltzmann- Gibbs entropy : • Is always positive • is expansible • is additive • L. Boltzmann • For two probabilistic events A and B where p(A+B)=p(A)p(B): • S(A+B) = S(A) + S(B)

23. The NESP approach and Tsallis entropy • Tsallis entropy’s properties: • expansibility • non-negativity • non-additivity • C. Tsallis

24. Tsallis entropy is non-additive • For two probabilistic events A and B where p(A+B)=p(A)p(B) : • q<1 supperadditivity • q>1 subadditivity • q=1 additivity ( Boltzmann-Gibbs )

25. Optimizing Tsallis entropy Sq • For the probability distribution p(X) of the continuous variable X, Tsallis entropy Sq is given by the integrated formulation as follows: • The second constraint is the condition about the generalized expectation value (q-expectation value), Xq defined as: • where Pq(X) is the escort probability given (Tsallis, 2009) as follows:

26. the probability distribution p(X) of the seismic parameter X (i.e., the seismic moment M, the inter-event time τ or the inter-event distance D) is obtained by maximizing the non extensive entropy under appropriate constraints with the Lagrange multipliers method. Herein we use the normalization condition and a generalized expectation value definition as appropriate parameters to optimize the non-extensive entropy • . We note that Xq denotes the generalized expectation value (q-expectation value) and Pq(X) is the escort probability (Tsallis, 2009). The Lagrange multipliers method • the q-partition function

27. the cumulative distribution function is given by the expression . P(>X) = expq(-BX),

28. The q-exponential distribution consists a generalization of the Zipf-Mandelbrot distribution (Mandelbrot, 1983), where the standard Zipf-Mandelbrot distribution corresponds to the case q>1 (Abe and Suzuki, 2003). • In the limit q→1 the q-exponential lead to the ordinary exponential function. • If q>1 equation q-exponential exhibits an asymptotic power-law behavior with slope -1/(q-1). • In contrast, for 0<q<1 a cut-off appears (Abe and Suzuki 2003, 2005).

29. Theq-exponential distribution for various values of q and for X0=1 in log-linear (a) and log-log scales (b). • The distribution is convex for q> 1 and concave for q< 1. • For q< 1, it has a vertical asymptote at x = (1-q)-1 and for q> 1 an asymptotic slope -1/(q-1). For q=1 the standard exponential distribution is recovered.

30. Another type of distributions that are deeply connected to statistical physics is that of the squared variable X2. • In BG statistical physics, the distribution of X2 corresponds to the well-known Gaussian distribution. • If we optimize Sqfor X2, we obtain a generalization of the normal Gaussian that is known as • q-Gaussian distribution and has the form: • . • In the limit q→1 recovers the normal Gaussian distribution. For q> 1, the q-Gaussian distribution has power-law tails with slope -2/(q-1), thus enhancing the probability of the extreme values.

31. The q-Gaussian distribution for various values of q and for p0=0.5 and X0=1 in linear (a) and log-linear scales (b). For q=1 the normal Gaussian distribution is recovered.

32. Experimental evidence of a non-extensive statistical physics behavior of fracture in triaxially deformed Etna basalt using acoustic emissions.

33. The 134s continuous AE record of the strain softening and failure part of the test in Etna basalt deformed at 40 MPa effective pressure. The strain softening between peak stress and dynamic failure noted with the A and B arrows. • Cumulative AE moment release in Etna basalt deformed at 40 MPa effective pressure

34. qM=1.82

36. AEs in Etna’s basalt are described by the q-value triplet (qM, qτ , qD)=(1.82, 1.34, 0.65). • We note that in the case of a simple system described by Boltzmann - Gibbs statistics the q triplet is given by (qM, qτ , qD) = (1, 1, 1). • We point out that the sum of qτ and qD indices of the distribution of the inter-event time and distance is qτ+qD≈2, similar with that observed in regional seismicity data both from Japan and California and verified numerically using the two dimensional Burridge-Knoppoff model. In non-extensive statistical mechanics quantities that are described with q (>0) and 2-q (>0) are defined as dual.

37. A non-extensive view of the Pressure Stimulated Current relaxation during repeated abrupt uniaxial load-unload in rock samples • Temporal recording of the PSC signal along with stress evolution during the first loading cycle in amphibolites samples.

40. SUPERSTATISTICAL APPROACH • Theoretical developments of the superstatistics concept include many complex systems, such as the atmospheric turbulence, the cosmic-ray statistics and hydroclimatic fluctuations and Earth physics  • In order to have a superstatistical approach to PSC relaxation, we assume a simple model which consists of a great number of relaxed subdomains defined by a fractures' network, where the local relaxation is given by that of a Debye exponential relaxation . • The expression denotes the relaxation of an individual subdomain when the local relaxation parameter β has a certain given value. Since PSCs are the observed macroscopic result of the relaxation of a great number of subdomains formed by a complex fracture's network, it is straightforward to assume that β is not a constant parameter, but an intensive parameter of the relaxation of the complex system that fluctuates on a relatively large scale.

41. Under this assumption the exponential model becomes superstatistical. If is distributed with probability density and fluctuates on a large time scale, then we can obtain the marginal distributions as a weighted average of a Debye exponential relaxation decay with respect to a distribution of relaxation time, leading to a relaxation function:

43. Non Extensive statistical physics approach to fault population distribution. Earthquake Fault System An earthquake fault system is a grouping of topologically complex faults or fault segments that have significant mutual interactions due to elastic or other stress transfer. The activity on the faults is strongly correlated and displays emergent space-time patterns that are properties of the system as a whole and not of the individual faults of which the system is composed.

44. Non Extensive statistical physics approach to fault population distribution. • Fault systems are among the most relevant paradigms of the so-called self-organized criticality (Bak et al., 1987) and represent a complex spatio–temporal phenomenon related to the deformation and sudden rupture of the earth‘s crust. • Crack and fault populations are characterized by scale-invariance so that their length distribution decays as a power-law: • where N is the number of faults with length equal or greater than L, A is a constant and D is the scaling exponent (Main, 1996; Turcotte, 1997).

45. Systematic variations in fault size distributions during fault system evolution have been predicted from physical experiments (Ishikawa and Otsuki, 1995) and numerical models (Cowie et al., 1995; Cladouhos and Marrett, 1996; Gupta and Scholz, 2000). • Some continental fault populations obey a power-law (fractal) distribution of size (e.g. Scholz and Mandelbrot 1989; Spyropoulos et al., 1999). On the other hand Cowie et al., (1994) showed that the size distribution of normal faults in oceanic crusts is best described by an exponential function.

46. Non Extensive statistical physics approach to fault population distribution. A case study from the Southern Hellenic Arc (Central Crete). • All faults in Central Crete graben. • The q=1.16

47. A non-extensive statistics of the fault-population at the Valles Marineris extensional province, Mars. • Explore using the principles of non extensive statistical mechanics the fault population statistics derived for an extraterrestrial data set selected in a well-studied planet as Mars is. • Applied techniques of digital image interpretation unequivocal evidence for normal faulting (Banerdt et al., 1992), thrust faulting (Watters, 1993) and strike-slip faulting (Schultz, 1989) on Mars has been demonstrated. Given that little or no liquid water is present at the Martian surface, along with the presence of only a tenuous atmosphere, erosion and deposition processes that can degrade fault scarps are extremely slow, promoting well-preserved structures over geologic timescales and facilitating the recognition, interpretation, and measurement of these relatively pristine structures in orbital images (Schultz et al., 2010) B) strongly interacting (or linked) fault segments • independent fault segments

48. B ) strongly interacting (or linked) fault segments • independent fault segments • Semi-q-log plot of the cumulative distribution function CDF of a) independent and b) strongly interacting (or linked) fault segments from Valles Marineris extensional province, Mars. The analysis of fault lengths indicates that q =1.10 for the independent faults , while q=1.75 for the linked faults,

49. A ) Compressional Mars faults b) Extensional Mars Faults • The normalized cumulative distribution function P(>L)for Mars (a) compressional and (b) extensional faults. The black live is the q-exponential fitting for a) qc=1.114 for the trust (compressional) faults and b) qe=1.277 for the normal (extensional ) ones (Vallianatos, 2013).