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Nonextensive models for earthquakes 6 0 0 R. Silva∗ 2 Observatório Nacional, Rua Gal. José Cristino 77, 20921-400 Rio de Janeiro - RJ,Brasil and Universidade do Estado do Rio Grande do Norte, 59610-210, Mossoró, RN, Brasil n a J G. S. França† and C. S. Vilar‡ 9 Departamento de F´ısica, Universidade Federal do R. G. do Norte, 59072-970, Natal, RN, Brasil ] J. S. Alcaniz§ h Observatório Nacional, Rua Gal. José Cristino 77, 20921-400 Rio de Janeiro - RJ,Brasil c e (Dated: February 2, 2008) m Wehaverevisitedthefragment-asperityinteractionmodelrecentlyintroducedbySotolongo-Costa - andPosadas (Physical ReviewLetters92, 048501, 2004) [1]byconsidering adifferentdefinition for t a meanvaluesinthecontextofTsallisnonextensivestatisticsandintroducinganewscalebetweenthe t earthquakeenergyandthesizeoffragmentǫ∝r3. Theenergydistributionfunction(EDF)deduced s . in our approach is considerably different from the one obtained in the above reference. We have at also tested the viability of this new EDF with data from two different catalogs (in three different m areas), namely, NEIC and Bulletin Seismic of the Revista Brasileira de Geof´ısica. Although both approaches providevery similar values for thenonextensiveparameter q, otherphysical quantities, - d e.g., theenergy density differsconsiderably, byseveral orders of magnitude. n o PACSnumbers: 89.75.Da;91.30.Bi;91.30.Dk c [ I. INTRODUCTION tionsandradonemissionofvolcanos[7]. Morerecently,a 2 very interesting model for earthquakes dynamics related v to Tsallis nonentensive framework has been proposedby 4 Over the last two decades, a great deal of atten- Sotolongo-Costa and Posadas (SCP Model) [1]. Such a 4 tion has been paid to the so-called nonextensive Tsallis 3 model consists basically of two rough profiles interact- entropy, both from theoretical and observational view- 1 ing via fragments filling the gap between them, where 1 points. This particular nonextensive formulation [2, 3] the fragments are produced by local breakage of the lo- 5 seems to present a consistent theoretical tool to investi- cal plates. By using the nonextensive formalism the au- 0 gate complex systems in their nonequilibrium stationary thorsofRef. [1]notonlyshowedthe influenceofthe size / states, systems with multifractal and self-similar struc- t distribution of fragments on the energy distribution of a tures, systems dominated by long-range interactions, earthquakes but also deduced a new energy distribution m anamolous diffusion phenomena, among others. Some function (EDF), which gives the well-known Gutenberg- d- recent applications of Tsallis entropy Sq6=1 to a number Richter law [8] as a particular case. of complex scenarios is now providing a more definite n picture of the kind of physical problems to which this However,indealingwiththisnonextensiveframework, o c q-formalism can in fact be applied. a particular attention must be paid to the possible def- : initions for mean values, which play a fundamental role v In this regard, systems of interest in geophysics has within the domain of this nonextensive statistics [9]. In i also been studied in light of this nonextensive formal- X this concern, recent studies of the properties of the rela- ism. In this particular context, the very first investiga- r tive entropy and the Shore-Johnson theorem for consis- a tionwasdoneby Abe [4]whoshowedthatthe statistical tent minimum cross-entropy principle, revealed the ne- properties of three-dimensionaldistance between succes- cessity of the so-called q-expectation value in studies in- siveearthquakesfollowaq-exponentialfunctionwiththe volving this nonextensive statistical mechanics (see [10] nonextensive parameter lying in the interval [0,1] [5]. for details). Thus, by introducing this q-definition of Since then, other geophysical analyses have been per- mean value we re-analyzed the fragment-asperity inter- formed as, for instance, the statistics of the calm time, action model of Sotolongo-Costaand Posadas[1]. More- which indicates a scale-free nature for earthquake phe- over,anewscalelawbetweenthereleasedrelativeenergy nomena and corresponds to a q-exponential distribution ǫ and the 3-dimensional size of fragments has also been with q > 1 [6], and models for temperature distribu- introduced. By using the standard method of entropy maximization we also deduced a new energy distribution function, which differs considerably from the one obtained in Ref. [1]. In order to test the viability of ∗Electronicaddress: [email protected],[email protected] our appoach we used data taken from two seismic cata- †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] logs, namely, NEIC and Bulletin Seismic of the Revista §Electronicaddress: [email protected] Brasileira de Geofisica. It is shown that although both 2 approachesprovide very similar values for the nonexten- also necessary since the process of violent fractioning is sive parameter q, the other physical quantity, e.g., the very probably a nonextensive phenomenon, leading to energy density differ by several orders of magnitude. long-range interactions among the parts of the object Thispaperisorganizedasfollows. InSec. II,thestan- being fragmented (see, e.g., [1, 13]). In reality, such an dard formalism of nonextensive statistical mechanics is influence was earlier emphasized in other investigations reexamined, as well as the theorical basis of SCP model. [14]. In general lines, the SCP model follows similar ar- In Sec III, a new EDF is analytically calculated through guments to those presented in Refs. [15] being, however, extremization of Tsallis’ entropy under the constrains of a more realistic seismic model than the one proposed in the q-expectation value and normalization condition. In Ref. [16]. Inparticular,thetheoreticalingredientsreads: Sec IV, we test this new EDF with data from two dif- • the mechanism of relative displacement of fault ferent catalogs and estimate the best-fit values for the plates is the main cause of earthquakes; nonextensive parameter q and the proportionality constant between the releasedrelative energy ǫ and the vol- • the surfacesofthe tectonic plates areirregularand 3 ume of the fragments r , i.e., the energy density, a. We the fragments filling the space between them are end this paper by emphasizing the main results in the very diverse and have irregular shapes; conclusion Section. • the mechanism of triggering earthquakes is estab- lished through the combination of irregularities of II. NON-EXTENSIVE FRAMEWORK AND SCP the fault planes and the distribution of fragments MODEL between them; • the fragment distribution function and conse- In this Section, we recall the nonextensive theoretical quently the EDF emerges naturally from a nonex- basis of the SCP model. As widely known, the Tsal- tensive framework. lis’ statistics generalizes the Botzmann-Gibbs statistics inwhatconcernsthe conceptofentropy. Suchformalism From the above arguments, the EDF deduced in Ref. is based on the parametric class of entropies given by [1] is given by Sq6=1 =−kBZ pq(σ)lnqp(σ)dσ, (1) log(N>m) = logN +(cid:18)12−−qq(cid:19)× (3) wherekB is the Boltzmannconstant. Inthe SCPmodel, ×log 1+a(q−1)(2−q)((q1−−q2)) ×102m . p(σ) stands for the probability of finding a fragment of h i relative surface σ (which is defined as a characteristic According to Ref. [1], the above expression describes surface of the system), q is the nonextensive parameter verywelltheenergydistributioninalldetectablerangeof and the q-logarithmic function above is defined by magnitudes, unlike the empirical formula of Gutenberg- ln p = (1−q)−1(p1−q−1), (p>0) (2) Richter [8]. q which recovers the standard Boltzmann-Gibbs entropy S1 = −kB plnpd3p in the limit q → 1. It is worth III. NEW APPROACH mentioningRthat most of the experimental evidence sup- portingTsallisproposalarerelatedtothepower-lawdis- Now,letus discuss the standardmethodofmaximiza- tribution associated with Sq6=1 descripition of the classi- tionoftheTsallisentropy. Hereandhereafter,theBoltz- cal N-body problem [11]. mann constant is set equal to unity for the sake of sim- The SCP model is a simple approach for earthquakes plicity. Thus, the functional entropy to be maximized is dynamics revealing a very interesting application of the Tsallis’ framework. Indeed, the fundamental idea consists in the fact that the space between faults is filled ∞ δS∗ =δ S +α p(σ)dσ−βσ =0, (4) with the residues of the breakage of the tectonic plates. q (cid:18) q Z0 q(cid:19) In this regard, the authors studied the influence of the size distribution of fragments on the energy distribution where α and β are the Lagrange multipliers. The con- of earthquakes. The theoretical motivation follows from strains used above are the normalizationof the distribu- the fragmentation phenomena [12] in the context of the tion geophysics systems. In this latter work, Englaman et al ∞ showed that the standard Botzmann-Gibbs formalism, p(σ)dσ =1 (5) althoughuseful,cannotaccountforanimportantfeature Z0 of fragmentation process, i.e., the presence of scaling in and the q-expectation value the size distribution of fragments, which is one of the main ingredients of the SCP approach. Thus, a nonex- ∞ σ =<σ > = σP (σ)dσ (6) tensive formalism is not only justified in SCP model but q q Z0 q 3 Samambaia Fault New Madrid Fault Anatolian Fault 100 100 q = 1.60; a = 1.3 x 1010 100 q = 1.63; a = 1.2 x 1010 q = 1.71; a = 2.8 x 1010 10-1 m)10-1 m) 10-1 m)10-2 G(> G(> G(> 10-3 10-2 10-2 10-4 (a) (b) (c) 10-5 3.0 3.5 4.0 4.5 5.0 5.5 3.0 3.5 4.0 4.5 5.0 5.5 3 4 5 6 7 8 m m m FIG. 1: The relative cumulative number of earthquakes [Eq. (12)] as a function of the magnitude m. In all panels, the data points correspond to earthquakes lying in the interval 3 < m < 8. (a): Samambaia fault - Brazil: 100 events from Bulletin Seismic of the Revista Brasileira de Geof´ısica. (b): New Madrid fault - USA: 173 data points taken from NEIC catalog. (c): Anatolian fault - Turkey: 8980 events from NEIC catalog. The best-fit values for the parameters q and a are shown in the respective Panel. A summary of this analysis is also shown in Table I. with the escort distribution [17] given by is straightforwardto show that Pq = ∞pq(σ) . (7) p(ε)dε= Cε−13dε , (9) 0 pq(σ)dσ 1+C′ε2/3 q−11 R By considering the same physical arguments of Ref. [1], whichhas alsoa power-la(cid:2)wformwit(cid:3)h C andC′ givenby wederive,aftersomealgebra,the followingexpresionfor the fragment size distribution function 2 (1−q) (1−q) 1−1q C = 3a2/3 and C′ =−(2−q)a2/3. (10) p(σ)= 1− (σ−σ ) , (8) (cid:20) (2−q) q (cid:21) Intheaboveexpression,theenergyprobabilityiswritten which corresponds to the area distribution for the frag- as p(ε) = n(ε)/N, where n(ε) corresponds to the num- mentsofthefaultplates. Here,however,differentlyfrom ber of earthquakes with energy ε and N total number Ref. [1], which assumesε∼r, we use a new energyscale earthquakes. ε ∼ r3. Thus, the proportionality between the released relativeenergyǫandr3 (risthesizeoffragments)isnow givenby σ−σ =(ε/a)2/3, where σ scaleswith r2 anda IV. TESTING THE NEW EDF WITH THE (the proportioqnality constant between ε and r3) has di- CUMULATIVE NUMBER OF EARTHQUAKES mension of volumetric energy density. In particular,this In order to test the viability of the new EDF above new scale is in accordance with the standard theory of derived[Eq. (9)] we introduce the cumulative number of seismic rupture, the well-known seismic moment scaling earthquakes,given by integral [1] with rupture length (see, for instance [18]). ThenewEDFofearthquakesis,therefore,obtainedby N ∞ ǫ> changing variables from σ−σq to (ε/a)2/3. From (8) it N =Zε p(ε)dε, (11) 4 (12), provide a very good fit to the experimental data TABLE I: Limits to q and a of the two catalogs here considered. It is worth empha- Fault Ref. q a sizing, however,that the energy density differ by several orders of magnitude from our model to the originalSCP California - USA [1] 1.65 5.73×10−6 model. Therefore, we expect that other independent es- Iberian Penisula - Spain [1] 1.64 3.37×10−6 timates ofthe parametera mayindicate whichapproach Andaluc´ıa - Spain [1] 1.60 3.0×10−6 is more physically realistic. The estimates of the parameters q and a obtained in this paper and in Ref. [1] are Samambaia - Brazil This Paper 1.60 1.3×1010 summarized in Table 1. NewMadrid - USA This Paper 1.63 1.2×1010 V. CONCLUSION Anatolian - Turkey This Paper 1.71 2.8×1010 InRef. [10],whatseemstobethecorrectdefinitionfor expectationvalueswithintheTsallisnonextensivestatis- where Nε> is the number of earthquakes with energy tical mechanics was rediscussed. Based on properties of larger than ε. Now, substituting (9) into (11), and con- the generalizedrelative entropies andthe Shore-Johnson 1 sidering m = 3logε (m stands for magnitude) it is pos- theorem, it was shown that the expectation value of any sible to calculate the above expression. In reality, note physical quantity in this extended framework converges thatdependingonthevalueofqthelimitsoftheintegral to the normalized q-expectation value, instead of to the (11) presents a cutoff on the maximum value allowedfor ordinary definition. energy ǫ, which is givenby ǫmax = a2/3(2−q)/(1−q) In this paper, by considering this necessity of q- for the intervals q < 1 and q > 2,pwhile for 1 < q < 2 expectationvaluesinTsallisnonextensiveframework,we the cutoff is absentin the distribution. Note alsothat in have revisited the fragment-asperity interaction model the limit q → 1, ǫmax → ∞ and p(ε) goes to the expo- for earthquakes,as introduced in Ref. [1]. A new energy nentialfunction. Asmatteroffact,thecalculationofthe distribution function has been calculated, which allowed integral (11) for q 6=1 leads to the general expression ustodeterminethe relativecumulativenumberofearth- quakes as a function of the magnitude. Additionally, a 2−q log(N ) = logN + × (12) new scale law between the releasedrelative energy ǫ and >m (cid:18)1−q(cid:19) thevolumeoffragmentsr3 hasalsobeenintroduced,i.e., 1−q 102m in agreement with the so-called seismic moment scaling ×log 1− × , (cid:20) (cid:18)2−q(cid:19) (cid:18) a2/3 (cid:19)(cid:21) with rupture length. As discussed earlier, although our analysis and the one presented in Ref. [1] provide very which, similarly to the modified Gutenberg-Ricther law similarvaluesforthenonextensiveparameterq,theother (See,e.g.,Refs. [19]formoredetails),describesappropri- physicalquantity,e.g.,theenergydensitydifferbyseveral ately the energy distribution in a wider detectable range orders of magnitude. It would be interesting, therefore, of magnitudes. if we could have experimental estimates for these quan- Figure 1 shows the relative cumulative number of tities in order to compare the predictions of the models. earthquakes(G =N /N) as a function of the mag- Finally,it is worthmentioning thatthe estimatesfor the m> m> nitudem. Thedatapoints,correspondingtoearthquakes nonextensive parameter from the two catalogs here con- events lying in the interval 3 < m < 8, were taken from sidered(Fig. 1)areconsistentwiththeupperlimitq <2, two different catalogs, namely, Bulletin Seismic of the obtained from several independent studies involving the Revista Brasileira de Geof´ısica (left panels) and NEIC Tsallis nonextensive framework [20]. (central and right panels). The left, central and right Acknowledgments: The authors thank the anonymous Panels show the results of our analysis for the Samam- refereesfortheirvaluablesuggestionsandcomments. We baia fault, Brazil (100 events), New madrid fault, USA also thank the partial support by the Conselho Nacional (173 events), and Anatolian fault, Turkey (8980 events), de Desenvolvimento Cient´ıfico e Tecnológico (CNPq - respectively. We note that, similarly to original version Brazil). JSAissupportedbyCNPq(307860/2004-3)and of SCP model, our approach, represented by Eqs. (8)- CNPq (475835/2004-2). CSV is supportedby FAPERN. [1] O. Sotolongo-Costa, and A. Posadas, Phys. Rev. Lett. by S. R. A. Salinas and C. Tsallis; Nonextensive En- 92, 048501 (2004). tropyInterdisciplinary Applications, edited by M. Gell- [2] C. Tsallis, J. Stat. Phys. 52, 479 (1988); See also MannandC.Tsallis(OxfordUniversityPress,NewYork, http://tsallis.cat.cbpf.br/biblio.htm for an updated bib- 2004). liography. [4] S. Abe, and N. Suzuki, J. Goephys. Res. 108 (B2), 2113 [3] Braz. J. Phys.29,, 1 (1999), Special Issueon Nonexten- (2003). sive Statistical Mechanics and Thermodynamic, edited [5] S. Abe, Y. 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