Pathway Model and Nonextensive Statistical Mechanics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t PATHWAY MODEL AND NONEXTENSIVESTATISTICAL MECHANICS
A.M. Mathai
Centre for Mathematical Sciences Pala CampusArunapuram P.O., Pala, Kerala-686574, IndiaandDepartment of Mathematics and Statistics, McGill UniversityMontreal, Canada H3A2K6
H.J. Haubold
Office for Outer Space Affairs, United NationsP.O. Box 500, A1400 Vienna, AustriaandCentre for Mathematical Sciences Pala CampusArunapuram P.O., Pala, Kerala-686574, India
C. Tsallis
Centro Brasileiro de Pesquisas Fisicasand National Institute of Science and Technology of Complex SystemsRua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, BrazilandSanta Fe Institute1399 Hyde Park Road, Santa Fe, NM 87501, USA
Abstract
The established technique of eliminating upper or lower parameters in a generalhypergeometric series is profitably exploited to create pathways among confluenthypergeometric functions, binomial functions, Bessel functions, and exponential se-ries. One such pathway, from the mathematical statistics point of view, resultsin distributions which naturally emerge within nonextensive statistical mechanicsand Beck-Cohen superstatistics, as pursued in generalizations of Boltzmann-Gibbsstatistics.
Email addresses: [email protected] (A.M. Mathai), [email protected] (H.J. Haubold), [email protected] (C. Tsallis).
Preprint submitted to Elsevier 16 December 2018
INTRODUCTION
The celebrated Boltzmann-Gibbs (BG) statistical mechanics, in its classicalversion, typically holds for many-body systems whose microscopic nonlineardynamics is ergodic , either in the entire Γ phase space, or in at least one of itssubspaces determined by relevant symmetry considerations. For example, forclassical ferromagnets (say the infinite-spin Heisenberg ferromagnet in threedimensions) exhibiting a second order phase transition, ergodicity applies tothe entire Γ phase space for temperatures above the critical one, and only toone of the subspaces generated by the corresponding breakdown of symmetryfor temperatures below the critical one. Analogous requirements must be sat-isfied for quantum systems, where the role of the Γ phase space is played bythe appropriate Hilbert or Fock spaces. A fundamental question arises for theplethora of physical systems which violate ergodicity in the sense just men-tioned:
Is it possible to have for them a statistical mechanical theory similarto the usual one, and also connected to thermodynamics?
It was suggested in 1988 [1] that this is indeed possible based on a simplehypothesis, namely the generalization of the BG entropy, given (say in itscontinuous version) by S BG = − k Z dx p ( x ) ln p ( x ) . (1)The generalization that was then proposed is given by S q = k − R dx [ p ( x )] q q − q ∈ R ; S = S BG ) . (2)Before proceeding, let us mention here that entropic forms generalizing theBoltzmann-Gibbs-Shannon-von Neumann one have in fact a long history ininformation theory, cibernetics and related areas [2,3]. Indeed, along the years,the same or similar or related forms have been introduced again and againas possible mathematical functionals. For example, the Renyi form (definedhere below) has been useful as a characterization of multifractal geometry. Itappears, however, to be inadequate for thermodynamics since it is not con-cave for an important range of its parameter α , namely for α >
1, wheremany physical systems exist. Although independently postulated, the entropicfunctional S q turns out to be mathematically very close to those of Havrda-Charvat, Daroczy, Lindhard-Nielsen, and Mathai-Rathie . In physics, this It is by no means rare that abstract ideas formulated in some scientific areaeventually find interesting applications in other areas. A celebrated such exampleis the normal distribution. It was first introduced by Abraham De Moivre in 1733, S q under appropriate constraints (nonvanishing firstmoment, or nonvanishing second moment if the first moment is zero) yieldsthe q -exponential form [ p ( x ) ∝ e − β xq , or p ( x ) ∝ e − β x q respectively], where e zq ≡ [1 + (1 − q ) z ] / (1 − q )+ , being [ u ] + = u if u >
0, and zero otherwise; e z = e z .These functions belong to a complex net of related and more general functions,whose systematic discussion constitutes the aim of the present paper.The above q -exponential functions emerge in a considerable amount of natural,artificial and social systems. For example (i) The velocity distribution of (cellsof) Hydra viridissima follows a q = 3 / Dictyostelium discoideum follows a q = 5 / q = 2 PDF in the starved state [7]; (iii) The velocity distributionin defect turbulence [8]; (iv) The velocity distribution of cold atoms in a dis-sipative optical lattice [9]; (v) The velocity distribution during silo drainage[10,11]; (vi) The velocity distribution in a driven-dissipative 2D dusty plasma,with q = 1 . ± .
01 and q = 1 . ± .
01 at temperatures of 30000 K and61000 K respectively [12]; (vii) The spatial (Monte Carlo) distributions of atrapped Ba + ion cooled by various classical buffer gases at 300 K [13]; (viii)The distributions of price returns and stock volumes at the stock exchange,as well as the volatility smile [14,15,16,17]; (ix) The distributions of returns ofmagnetic field fluctuations in the solar wind plasma as observed in data fromVoyager 1 [18] and from Voyager 2 [19]; (x) The distributions of returns inthe Ehrenfest’s dog-flea model [20,21]; (xi)The distributions of returns in thecoherent noise model [22]; (xii) The distributions of returns of the avalanchesizes in the self-organized critical Olami-Feder-Christensen model, as well asin real earthquakes [23]; (xiii) The distributions of angles in the HM F model[24]; (xiv) The distribution of stellar rotational velocities in the Pleiades [25];(xv) The relaxation in various paradigmatic spin-glass substances throughneutron spin echo experiments [26]; (xvi) Various properties directly relatedwith the time dependence of the width of the ozone layer around the Earth[27]; (xvii) The distribution of transverse momenta in high energy collisions ofelectron-positron, proton-proton, and heavy nuclei (e.g., Pb-Pb and Au-Au)[28,29,30,31,32,33,34], the flux of solar neutrinos [35], and the energy distri-bution of cosmic rays [36]; (xviii) Various properties for conservative and dis-sipative nonlinear dynamical systems [37,38,39,40,41,42,43,44,45]; (xix) Thedegree distribution of (asymptotically) scale-free networks [46,47], and others. then by Pierre Simon Laplace in 1774, then by Robert Adrain in 1808, and finally byCarl Friedrich Gauss in 1809, who connected the normal distribution to the theoryof errors, applicable in all experimental sciences. q -exponential functions with other functions (derivable ornot from various entropic forms) within a variety of pathways, some of whichalso emerge in applications. This leads us to the next Section. Consider a confluent hypergeometric series F ( a ; b ; x ) = ∞ X r =0 ( a ) r ( b ) r x r r ! , ( c ) r = c ( c + 1) ... ( c + r − , ( c ) = 1 , c = 0 . (3)If we want to remove the denominator parameter, then a well known techniquein the area of special functions is to replace x by b x and then take the limitwhen b → ∞ . Due to the fact thatlim b →∞ b r ( b ) r = lim b →∞ b r b ( b + 1) ... ( b + r −
1) = 1 , (4)we havelim b →∞ F ( a ; b ; bx ) = F ( a ; ; x ) = (1 − x ) − a , | x | < . (5)Hence a pathway between the binomial function (1 − x ) − a and the F seriesis given by the limit of b r ( b ) r when b → ∞ . Going the other way one can buildup a bridge between F and F by introducing b r ( b ) r into a F series. That is, ∞ X r =0 ( a ) r x r r ! ≈ ∞ X r =0 ( a ) r ( b ) r ( bx ) r r ! = F ( a ; b ; bx ) f or large b. (6)Similarly one can go back and forth from a Bessel function F to a F orto a F which is the exponential series. Let us look at going from a binomialseries to an exponential series.(1 − x ) − a = ∞ X r =0 ( a ) r x r r ! ⇒ lim a →∞ ( a ) r r ! ( xa ) r = ∞ X r =0 x r r ! = e x . (7)In other words,e − cx = lim a →∞ ∞ X r =0 ( − a ) r r ! ( cxa ) r = lim a →∞ (1 − cxa ) a = lim α → [1 − c (1 − α ) x ] − α . (8)4hus a pathway between the exponential function e − cx , c > − c (1 − α ) x ] − α can be created with the help of the pathwayparameter α . When α is very close to 1, the binomial and exponential func-tions are very close to each other and they will be farther apart when α isaway from 1. Observe that e − cx , c > , < x < ∞ and [1 − c (1 − α ) x ] − α ,0 < x < c (1 − α ) , c > , α < c ( α − x ] − α − , c > , α > , x > f ( x ) = λ x γ e − cx , c > , x > f ( x ) = λ x γ [1 − c (1 − α ) x ] − α , c > , α < , < x < c (1 − α ) ; f ( x ) = λ x γ [1 + c ( α − x ] − α − , α > , c > , x > , (9)where λ , λ , λ are the appropriate normalizing constants, can be created withthe help of the pathway parameter α . Observe that in f , f and f one canreplace x by | x | , −∞ < x < ∞ or x by | x | δ , δ > f stays in the exponential/gamma type den-sities, f stays as a type-1 beta form and f a type-2 beta form. By exploitingthese observations, Mathai has introduced [48] the pathway model connectingexponential type and binomial type functions.Another rich area is the class of Bessel functions. As indicated above, a Besselfunction can be written in terms of a hypergeometric function F ( ; b ; x ) andone can remove the denominator parameter b by replacing x by bx and thenusing the limit b → ∞ . In other words,lim b →∞ F ( ; b ; − bx ) = F ( ; ; − x ) = e − x = lim α → F ( ; 11 − α ; − x − α ) . (10)Thus α can provide a pathway between Bessel functions and exponential func-tions. If the exponential form gives the stable situation, then the parameter α will provide a pathway between stable and chaotic situations. So far thisarea is not explored. In this connection one can obtain an interesting resultby using the integral representation of a Gauss hypergeometric function F ,namely, F ( a, b ; c ; − z ) = Γ( c )Γ( a )Γ( c − a ) Z x a − (1 − x ) c − a − (1 + zx ) − b d x, (11) ℜ ( a ) > , ℜ ( c − a ) > , | z | < . F ( a ; c ; − z ) = lim b →∞ F ( a, b ; c ; − xb ) , | x | <
1= Γ( c )Γ( a )Γ( c − a ) lim b →∞ Z x a − (1 − x ) c − a − (1 + zxb ) − b d x, | z | <
1= Γ( c )Γ( a )Γ( c − a ) Z x a − (1 − x ) c − a − e − zx d x, | z | < . (12)Thus a pathway between F and F is given by (12). Many such results canbe obtained by using this technique of eliminating one or more numerator ordenominator parameters from a general hypergeometric series p F q .Thus for a real scalar random variable x , the pathway density can be writtenin the following form: f ( x ) = λ | x | γ [1 − a (1 − α ) | x | ] − α , a > , > a (1 − α ) | x | , α < . (13)A more general form of the pathway density is the following: f ( x ) = λ | x | γ [1 − a (1 − α ) | x | δ ] η − α , (14)where a > η, δ, γ, α ) are such that f ( x ) is normalizable. A large num-ber of commonly used statistical densities can be seen to be particular cases of(14), details may be seen in [48,49,50]. From the point of view of mathemat-ical statistics, nonextensive statistics [1,4,5,52,53,54] with constant density ofstates is a particular case of (13) for γ = 0 , x >
0. The case γ = 0 can be seen asthe particular case when the density of states is given by a power law (which isquite frequent in many physical systems). One of the forms of the Beck-Cohensuperstatistics [55,56] is a special case of (13) for γ = 0 , α > , x > In situations when an appropriate density is selected, one guiding princi-ple is the maximization of entropy. Entropy or a measure of uncertainty ina scheme or “information” in a scheme is traditionally measured by Shan-non entropy. Consider a discrete distribution P ′ = ( p , ..., p k ) , p i > , i =1 , ..., k, p + ... + p k = 1. This may also be looked upon as the sample space or6he sure event S is partitioned into mutually exclusive and totally exhaustiveevents A , ..., A k , A ∪ ... ∪ A k = S, A i ∩ A j = O for all i and j , i = j with theprobability of the event A i , denoted by p i = P r ( A i ) , i = 1 , ..., k . If any p i isallowed to take the value zero also, then p i ≥ , i = 1 , ..., k . Shannon entropyon this scheme is S ( P ), where S ( P ) = − k X i =1 p i ln p i . (15)When a p i = 0, p i ln p i is to be interpreted as zero. Several characterizationtheorems on S ( P ) or axiomatic definitions may be seen from [3]. There areseveral extensions or generalizations of the measure S k ( P ). Classical general-izations in information theory are the Havrda-Charv´at measure H k,α ( P ), andthe R´enyi measure R k,α ( P ), where H α ( P ) = P ki =1 p αi − − α − , R α ( P ) = ln( P ki =1 p αi )1 − α , α = 1 , α > . (16)These are generalizations in the sense that when α → , H α ( P ) → S ( P )and R α ( P ) → S ( P ). Out of these, S ( P ) and R α ( P ) are additive and H α ( P ) isnonadditive. The additivity property is defined as follows: Consider a bivariatediscrete distribution in the sense p ij > , i = 1 , ..., m, j = 1 , ..., n such that P mi =1 P nj =1 p ij = 1. What happens if there is the product probability property(PPP), which in statistical literature is known as statistical independence.What happens is that p ij = p i q j , p + .. + p m = 1 , q + ... + q n = 1 or there isthe product probability property. When PPP holds, if the entropy in the jointdistribution ( P, Q ) = ( p ij ) , i = 1 , ..., m, j = 1 , ..., n is the sum of the entropieson P and Q then we say that there is additivity. It is easily seen that there isadditivity in S ( P ) and R α ( P ), that is, R α ( P, Q ) = R α ( P ) + R α ( Q ) and S ( P, Q ) = S ( P ) + S ( Q ) . (17)This additivity holds due to the logarithmic nature of the function in S k ( P )and R α ( P ) and the logarithm of a product of positive quantities being thesum of the logarithms. It is explained in [49] that logarithmic function entersinto an entropy measure due to the recursivity axiom which leads into a log-arithmic function necessarily.In the following we will concentrate on the q -type of generalization of entropymeasures, and review, for completeness, how the extremization of generalizedentropies yields the probability density which correspond to stationary states.7t was postulated [1] the entropy S α ( P ) = P ki =1 p αi − − α , α = 1 , α > . (18)To avoid confusion, let us mention that, in most of the literature of nonexten-sive statistical mechanics, the index α is noted q , and the entropy S α is noted S q . The Shannon form is obtained as the q ≡ α → H α ( P ), namely (2 − α − − α ). In the continuous case, the nonadditive entropy uponwhich nonextensive statistical mechanics is built is then, S α ( f ) = R x [ f ( x )] α d x − − α , α = 1 , α > . (19)Over all functions f , what is that particular f which will optimize the nonad-ditive entropy in (19)? If calculus of variation principle is used, then the Eulerequation for optimizing the entropy S α under the restrictions Z x f ( x )d x = 1 and Z x xf ( x )d x = E ( x ) = f ixed , f ( x ) ≥ , ∀ x (20)will yield the equation, ∂∂f [ f α − λ f − λ xf ] = 0 ⇒ f = [ λ + λ x ] α − = λ [1 + ( α − x ] α − (21)by taking λ λ = α − λ α − = λ , where λ and λ are Lagrangian mul-tipliers. The quantity λ can act as the normalizing constant. The condition E ( x ) = fixed, where E denotes the expected value, can be interpreted as theprinciple of conservation of the quantity x . When α → , f = λ e − x which isan exponential function. The derivation in (21) does not yield nonextensivestatistics in its most convenient form. But (21) gives an exponential functionwhen α → q -exponential function. In order to circumvent somedifficulties, it was replaced ([1,5]) the second condition that E ( x ) is fixed byfixing the expected value in the escort distribution. The escort density is givenby g ( x ) = f α ( x ) R x f α ( x )d x (22)8nd then nonextensive statistics has the form f = λ [1 − (1 − α ) x ] − α . (23)This form can produce densities for α < , α > α → x ( fλ ) = − ( fλ ) α . (24)One can introduce a general measure of entropy, which in the discrete andcontinuous cases are denoted by M α ( P ) and M α ( f ) respectively, where M α ( P ) = P ki =1 p − αi − α − , α = 1 , α < , M α ( f ) = R x [ f ( x )] − α d x − α − , (25) α = 1 , α <
2. A characterization of M α ( P ) is given in [3] (see also [49,50,51]).If M α ( f ) is optimized under the conditions that E ( x ) = fixed and that f ( x )is a density, then the Euler equation becomes ∂∂f [ f − α − λ f + λ xf ] = 0 ⇒ f = λ [1 − a (1 − α ) x ] − α , (26)where a > , − a (1 − α ) x > , λ λ is taken as a (1 − α ) with a > (cid:16) λ − α (cid:17) − α is taken as λ . Observe that (26) readily gives densities for α < , α > α →
1. Further, the entropy itself can be expressed as M α ( P ) = E ( p − α ) − α − , M α ( f ) = E [ f − α ( x )] − α − − α ) can be interpreted as the strength of information in f and thisexpected value is also associated with Kerridge’s “inaccuracy” measure. Asa simple mathematical remark, let us mention that if the entropy in (25) isoptimized in an ad hoc manner, namely that for all f ( x ) such that f ( x ) ≥ x > R x f ( x )d x < ∞ , R x [ x γ (1 − α ) ] f ( x )d x = fixed and R x [ x γ (1 − α )+ δ ] f ( x )d x =fixed, then we end up with the density f ( x ) = λx γ [1 − a (1 − α ) x δ ] − α , −∞ < α < , a > , (28)and λ is the normalizing constant. Through trivial changes in the notation,this expression recovers that of (14). 9s already mentioned, for γ = 0 , δ = 1 in (28) one has a particular case ofnonextensive statistics. For a > , α > α <
1, (28) gives a generalizedtype-1 beta form for 0 < x δ < a (1 − α ) , and for α >
1, (28) gives a generalizedtype-2 beta form. Superstatistics can produce only the type-2 beta form andnot the type-1 beta form.
We utilize the established technique of eliminating upper or lower parame-ters in a general hypergeometric series to create pathways among confluenthypergeometric functions, binomial functions, Bessel functions, and exponen-tial series. Mathai’s pathway, from the mathematical statistics point of view,results in distributions which also emerge within nonextensive statistics andBeck-Cohen superstatistics, pursued as generalizations of Boltzmann-Gibbsstatistics. It was shown that this pathway model can also be derived by opti-mizing a generalized entropic measure. Through Mathai’s pathway approach,exponential and binomial type functions are connected through the pathwaymodel parameter. The same pathway model also leads to a link between Besselfunctions and exponential functions. The pathway model covers statistical den-sities emanating in nonextensive statistics and Beck-Cohen superstatistics asspecial cases of (28). Related results are obtained by optimizing a generalmeasure of entropy in (25) (see also [1,5,51]). An open problem is identifiedthat would allow to entropically derive a general density of the form (28)within physically meaningful circumstances. Summarizing, relations betweenMathai’s pathway model and nonextensive statistics and Beck-Cohen super-statistics were exhibited.
Acknowledgments
A.M.M. and H.J.H. would like to thank the Department of Science and Tech-nology, Government of India for the financial assistance for this work undergrant No. SR/S4/MS:287/05. C.T. acknowledges partial support from CNPqand Faperj (Brazilian Agencies).
References [1] C. Tsallis,
Possible generalizations of Boltzmann-Gibbs statistics , J. Stat. Phys. , 479 (1988).
2] J. Havrda and F. Charvat, Kybernetika , 30 (1967); I. Vajda, Kybernetika ,105 (1968) [in Czeck]; Z. Daroczy, Inf. Control , 36 (1970); J. Lindhard and V.Nielsen, Studies in statistical mechanics , Det Kongelige Danske VidenskabernesSelskab Matematisk-fysiske Meddelelser (Denmark) (9), 1-42 (1971); B.D.Sharma and D.P. Mittal, J. Math. Sci. , 28 (1975); J. Aczel and Z. Daroczy[ On Measures of Information and Their Characterization , in
Mathematics inScience and Engineering , ed. R. Bellman (Academic Press, New York, 1975)];A. Wehrl, Rev. Mod. Phys. , 221 (1978); A. Renyi, in Proceedings of theFourth Berkeley Symposium , , 547 (University of California Press, Berkeley,Los Angeles, 1961); A. Renyi, Probability theory (North-Holland, Amsterdam,1970).[3] A.M. Mathai and P.N. Rathie,
Basic Concepts in Information Theory andStatistics: Axiomatic Foundations and Applications (Wiley Halsted, New York,and Wiley Eastern, New Delhi, 1975).[4] E.M.F. Curado and C. Tsallis,
Generalized statistical mechanics: connectionwith thermodynamics , J. Phys. A , L69 (1991); Corrigenda: , 3187 (1991)and , 1019 (1992).[5] C. Tsallis, R.S. Mendes and A.R. Plastino, The role of constraints withingeneralized nonextensive statistics , Physica A , 534 (1998).[6] A. Upadhyaya, J.-P. Rieu, J.A. Glazier and Y. Sawada,
Anomalous diffusion andnon-Gaussian velocity distribution of Hydra cells in cellular aggregates , PhysicaA , 549 (2001).[7] A.M. Reynolds,
Can spontaneous cell movements be modelled as L´evy walks? ,Physica A , 273 (2010).[8] K.E. Daniels, C. Beck and E. Bodenschatz,
Defect turbulence and generalizedstatistical mechanics , Physica D , 208 (2004).[9] P. Douglas, S. Bergamini and F. Renzoni,
Tunable Tsallis distributions indissipative optical lattices , Phys. Rev. Lett. , 110601 (2006); G.B. Bagci andU. Tirnakli, Self-organization in dissipative optical lattices , Chaos , 033113(2009).[10] R. Arevalo, A. Garcimartin and D. Maza, Anomalous diffusion in silo drainage ,Eur. Phys. J. E , 191-198 (2007)[DOI10.1140/epje/i2006-10174-1].[11] R. Arevalo, A. Garcimartin and D. Maza, A non-standard statistical approachto the silo discharge , in
Complex Systems - New Trends and Expectations , eds.H.S. Wio, M.A. Rodriguez and L. Pesquera, Eur. Phys. J.-Special Topics (2007) [DOI: 10.1140/epjst/e2007-00087-9].[12] B. Liu and J. Goree,
Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma , Phys. Rev. Lett. , 055003 (2008).[13] R.G. DeVoe,
Power-law distributions for a trapped ion interacting with aclassical buffer gas , Phys. Rev. Lett. , 063001 (2009).
14] L. Borland,
Closed form option pricing formulas based on a non-Gaussian stockprice model with statistical feedback , Phys. Rev. Lett. , 098701 (2002).[15] L. Borland, A theory of non-gaussian option pricing , Quantitative Finance ,415 (2002).[16] R. Osorio, L. Borland and C. Tsallis, Distributions of high-frequency stock-market observables , in
Nonextensive Entropy - Interdisciplinary Applications ,eds. M. Gell-Mann and C. Tsallis (Oxford University Press, New York, 2005).[17] S.M.D. Queiros,
On non-Gaussianity and dependence in financial in time series:A nonextensive approach , Quant. Finance , 475 (2005).[18] L.F. Burlaga and A.F.-Vinas, Triangle for the entropic index q of non-extensivestatistical mechanics observed by Voyager 1 in the distant heliosphere , PhysicaA , 375 (2005).[19] L.F. Burlaga and N.F. Ness, Compressible “turbulence” observed in theheliosheath by Voyager 2 , Astrophys. J. , 311 (2009).[20] B. Bakar and U. Tirnakli,
Analysis of self-organized criticality in Ehrenfest’sdog-flea model , Phys. Rev. E , 040103(R) (2009).[21] B. Bakar and U. Tirnakli, Return distributions in dog-flea model revisited ,Physica A , 3382 (2010).[22] A. Celikoglu, U. Tirnakli and S.M.D. Queiros,
Analysis of return distributionsin the coherent noise model , Phys. Rev. E , 021124 (2010).[23] F. Caruso, A. Pluchino, V. Latora, S. Vinciguerra and A. Rapisarda, Analysisof self-organized criticality in the Olami-Feder-Christensen model and in realearthquakes , Phys. Rev. E , 055101(R)(2007).[24] L.G. Moyano and C. Anteneodo, Diffusive anomalies in a long-rangeHamiltonian system , Phys. Rev. E , 021118 (2006).[25] J.C. Carvalho, R. Silva, J.D. do Nascimento and J.R. de Medeiros, Power lawstatistics and stellar rotational velocities in the Pleiades , Europhys. Lett. ,59001 (2008).[26] R.M. Pickup, R. Cywinski, C. Pappas, B. Farago and P. Fouquet, Generalizedspin glass relaxation , Phys. Rev. Lett. , 097202 (2009).[27] G.L. Ferri, M.F. Reynoso Savio and A. Plastino,
Tsallis q -triplet and the ozonelayer , Physica A , 1829 (2010).[28] I. Bediaga, E.M.F. Curado and J. Miranda, A nonextensive thermodynamicalequilibrium approach in e + e − → hadrons , Physica A , 156 (2000).[29] G. Wilk and Z. Wlodarczyk, Power laws in elementary and heavy-ion collisions- A story of fluctuations and nonextensivity? , Eur. Phys. J. A , 299 (2009).[30] T.S. Biro, G. Purcsel and K. Urmossy, Non-extensive approach to quark matter ,in
Statistical Power-Law Tails in High Energy Phenomena , Eur. Phys. J. A ,325 (2009).
31] V. Khachatryan et al (CMS Collaboration),
Transverse-momentum andpseudorapidity distributions of charged hadrons in pp collisions at √ s = 0 . and . T eV , J. High Energy Phys. , 041 (2010).[32] V. Khachatryan et al (CMS Collaboration), Transverse-momentum andpseudorapidity distributions of charged hadrons in pp collisions at √ s = 7 T eV ,Phys. Rev. Lett. , 022002 (2010).[33] Adare et al (PHENIX Collaboration),
Measurement of neutral mesons in p + p collisions at √ s = 200 GeV and scaling properties of hadron production ,preprint (2010), 1005.3674 [hep-ex].[34] M. Shao, L. Yi, Z.B. Tang, H.F. Chen, C. Li and Z.B. Xu,
Examination of thespecies and beam energy dependence of particle spectra using Tsallis statistics ,J. Phys. G (8), 085104 (2010).[35] G. Kaniadakis, A. Lavagno and P. Quarati, Generalized statistics and solarneutrinos , Phys. Lett. B , 308 (1996).[36] C. Tsallis, J.C. Anjos and E.P. Borges,
Fluxes of cosmic rays: A delicatelybalanced stationary state , Phys. Lett. A , 372 (2003).[37] M.L. Lyra and C. Tsallis,
Nonextensivity and multifractality in low-dimensionaldissipative systems , Phys. Rev. Lett. , 53 (1998).[38] E.P. Borges, C. Tsallis, G.F.J. Ananos and P.M.C. Oliveira, Nonequilibriumprobabilistic dynamics at the logistic map edge of chaos , Phys. Rev. Lett. ,254103 (2002).[39] G.F.J. Ananos and C. Tsallis, Ensemble averages and nonextensivity at the edgeof chaos of one-dimensional maps , Phys. Rev. Lett. , 020601 (2004).[40] F. Baldovin and A. Robledo, Nonextensive Pesin identity. Exact renormalizat-ion group analytical results for the dynamics at the edge of chaos of the logisticmap , Phys. Rev. E , 045202(R) (2004).[41] E. Mayoral and A. Robledo, Tsallis’ q index and Mori’s q phase transitions atedge of chaos , Phys. Rev. E , 026209 (2005).[42] A. Pluchino, A. Rapisarda and C. Tsallis, Nonergodicity and central limitbehavior in long-range Hamiltonians , Europhys. Lett. , 26002 (2007).[43] A. Pluchino, A. Rapisarda and C. Tsallis, A closer look at the indications of q -generalized Central Limit Theorem behavior in quasi-stationary states of theHMF model , Physica A , 3121 (2008).[44] G. Miritello, A. Pluchino and A. Rapisarda, Central limit behavior in theKuramoto model at the ’edge of chaos’ , Physica A , 4818 (2009).[45] M. Leo, R.A. Leo and P. Tempesta,
Thermostatistics in the neighborhood ofthe π -mode solution for the Fermi-Pasta-Ulam β system: From weak to strongchaos , J. Stat. Mech. P04021 (2010).
46] D.R. White, N. Kejzar, C. Tsallis,D. Farmer and S. White,
A generative modelfor feedback networks , Phys. Rev. E , 016119 (2006).[47] S. Thurner, F. Kyriakopoulos and C. Tsallis, Unified model for networkdynamics exhibiting nonextensive statistics , Phys. Rev. E , 036111 (2007).[48] A.M. Mathai, A Pathway to matrix variate gamma and normal densities , LinearAlgebra and Its Applications , 317 (2005).[49] A.M. Mathai and H.J. Haubold,
Pathway model, superstatistics, Tsallisstatistics and a generalized measure of entropy , Physica A , 110 (2007).[50] A.M. Mathai and H.J. Haubold,
On generalized entropy measures and pathways ,Physica A , 493 (2007).[51] G.L. Ferri, S. Martinez and A. Plastino,
The role of constraints in Tsallis’nonextensive treatment revisited , Physica A , 205 (2005).[52] M. Gell-Mann and C. Tsallis, Eds.,
Nonextensive Entropy: InterdisciplinaryApplications , (Oxford University Press, New York, 2004).[53] C. Tsallis,
What should a statistical mechanics satisfy to reflect nature? , PhysicaD , 3 (2004).[54] C. Tsallis,
Introduction to Nonextensive Statistical Mechanics - Approaching aComplex World , (Springer, New York, 2009).[55] C. Beck and E.G.D. Cohen,
Superstatistics , Physica A , 267 (2003).[56] C. Beck,
Stretched exponentials from superstatistics , Physica A , 96 (2006)., 96 (2006).