Nonadditive entropy and nonextensive statistical mechanics - An overview after 20 years

Brazilian Journal of Physics

Volumn 39, Issue 2A, 2009, Pages 337-356

  Tsallis, Constantino ^a,b  

^a CENTRO BRASILEIRO DE PESQUISAS FÍSICAS   (Brazil)

^b SANTA FE INSTITUTE   (United States)

Author keywords

Complex systems; Nonadditive entropy; Nonextensive statistical mechanics; Nonlinear dynamics

Indexed keywords

EID: 73349138004     PISSN: 01039733     EISSN: None     Source Type: Journal    
DOI: 10.1590/S0103-97332009000400002     Document Type: Article

References (137)

1. M. Planck, Uber irreversible Strahlugsforgange, Ann d. Phys. (4) (1900) 1, S. 69-122 (see [8]).
2. M. Planck, Verhandlungen der Deutschen Physikalischen Gessellschaft 2, 202 and 237 (1900) (English translation: D. ter Haar, S. G. Brush, Planck's Original Papers in Quantum Physics (Taylor and Francis, London, 1972)].
3. L.B. Okun, The fundamental constants of physics, Sov. Phys. Usp. 34, 818 (1991).
4. G. Cohen-Tannoudji, Les Constantes Universelles, Collection Pluriel (Hachette, Paris, 1998).
5. C. Tsallis and S. Abe, Advancing Faddeev: Math can deepen Physics understanding, Physics Today 51, 114 (1998).
6. L. Okun, Cube or hypercube of natural units?, hepth/0112339 (2001).
7. M.J. Duff, L.B. Okun and G. Veneziano, Trialogue on the number of fundamental constants, JHEP03 (2002) 023 [physics/0110060].
8. M.J. Duff, Comment on time-variation of fundamental constants, hep-th/0208093 (2004).
9. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479-487 (1988) [The proposal first came to my mind in August 1985, during a scientific event in Mexico City, and was further advanced in August 1987, during and immediately after a scientific event in Maceio-Brazil. But it was published only in 1988, particularly stimulated by the fruitful discussions held on the subject with E.M.F. Curado and H.J. Herrmann during the Maceio event].
10. C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer, New York, 2009).
11. A regularly updated Bibliography can be seen at http://tsallis.cat.cbpf.br/biblio.htm
12. C. Tsallis, R.S. Mendes and A.R. Plastino, The role of constraints within generalized nonextensive statistics, Physica A 261, 534 (1998).
13. E.M.F. Curado and C. Tsallis, Generalized statistical mechanics: connection with thermodynamics, J. Phys. A 24, L69 (1991)
14. E.M.F. Curado and C. Tsallis, Corrigenda: 24, 3187 (1991) and 25, 1019 (1992).
15. S. Abe, Instability of q-averages in nonextensive statistical mechanics, Europhys. Lett. 84, 60006 (2008).
16. R. Hanel, S. Thurner and C. Tsallis, On the robustness of q-expectation values and Rényi entropy, Europhysics Letters 85, 20005 (2009).
17. R. Hanel and S. Thurner, Stability criteria for q-expectation values, Phys. Lett. A 373, 1415 (2009).
18. S. Abe and G.B. Bagci, Necessity of q-expectation value in nonextensive statistical mechanics, Phys. Rev. E 71, 016139 (2005).
19. S. Abe, Why q-expectation values must be used in nonextensive statistical mechanics, Astrophys. Space Sci. 305, 241-245 (2006).
20. W. Thistleton, J.A. Marsh, K. Nelson and C. Tsallis, Generalized Box-Muller method for generating q-Gaussian random deviates, IEEE Transactions on Information Theory 53, 4805 (2007).
21. S. Umarov, C. Tsallis and S. Steinberg, On a q-central limit theorem consistent with nonextensive statistical mechanics, Milan J. Math. 76, 307 (2008).
22. J.-F. Bercher, Tsallis distribution as a standard maximum entropy solution with 'tail' constraint, Phys. Lett. A 372, 5657 (2008).
23. J. Naudts, The q-exponential family in statistical physics, Central Eur. J. Phys. 7, 405 (2009).
24. C. Tsallis, A.R. Plastino and R.F. Alvarez-Estrada, Escort mean values and the characterization of power-law-decaying probability densities, J. Math. Phys. 50, 043303 (2009).
25. V. Schwammle, F.D. Nobre and E.M.F. Curado, Consequences of the H-theorem from nonlinear Fokker-Planck equations, Phys. Rev. E 76, 041123 (2007).
26. V. Schwammle, E.M.F. Curado and F.D. Nobre, A general nonlinear Fokker-Planck equation and its associated entropy, Eur. Phys. J. B 58, 159 (2007).
27. S. Abe, Generalized molecular chaos hypothesis and Htheorem: Problem of constraints and amendment of nonextensive statistical mechanics, Phys. Rev. E 79, 041116 (2009).
28. G.L. Ferri, S. Martinez and A. Plastino, The role of constraints in Tsallis' nonextensive treatment revisited, Physica A 347, 205 (2005).
29. S.R.A. Salinas and C. Tsallis, eds., Nonextensive Statistical Mechanics and Thermodynamics, Braz. J. Phys. 29 (1) (1999).
30. S. Abe and Y. Okamoto, eds., Nonextensive Statistical Mechanics and Its Applications, Series Lecture Notes in Physics (Springer, Berlin, 2001).
31. P. Grigolini, C. Tsallis and B.J. West, eds., Classical and Quantum Complexity and Nonextensive Thermodynamics, Chaos, Solitons and Fractals 13, Issue 3 (2002).
32. G. Kaniadakis, M. Lissia and A. Rapisarda, eds., Non Extensive Thermodynamics and Physical Applications, Physica A 305, 1/2 (2002).
33. M. Sugiyama, ed., Nonadditive Entropy and Nonextensive Statistical Mechanics, Continuum Mechanics and Thermodynamics 16 (Springer-Verlag, Heidelberg, 2004).
34. H.L. Swinney and C. Tsallis, eds., Anomalous Distributions, Nonlinear Dynamics and Nonextensivity, Physica D 193, Issue 1-4 (2004).
35. M Gell-Mann and C. Tsallis, eds., Nonextensive Entropy - Interdisciplinary Applications (Oxford University Press, New York, 2004).
36. G. Kaniadakis and M. Lissia, eds., News and Expectations in Thermostatistics, Physica A 340, Issue 1-3 (2004).
37. H.J. Herrmann, M. Barbosa and E.M.F. Curado, eds., Trends and Perspectives in Extensive and Non-extensive Statistical Mechanics, Physica A 344, Issue 3/4 (2004).
38. C. Beck, G. Benedek, A. Rapisarda and C. Tsallis, eds., Complexity, Metastability and Nonextensivity (World Scientific, Singapore, 2005).
39. J.P. Boon and C. Tsallis, eds., Nonextensive Statistical Mechanics: New Trends, New Perspectives, Europhysics News 36, Number 6, 183-231 (EDP Sciences, Paris, 2005).
40. G. Kaniadakis, A. Carbone and M. Lissia, eds., News, Expectations and Trends in Statistical Physics, Physica A 365, Issue 1 (2006).
41. S. Abe, M. Sakagami and N. Suzuki, eds., Complexity and Nonextensivity - New Trends in Statistical Mechanics, Progress of Theoretical Physics Supplement 162 (Physical Society of Japan, 2006).
42. S. Abe, H. Herrmann, P. Quarati, A. Rapisarda and C. Tsallis, eds., Complexity, Metastability and Nonextensivity, American Institute of Physics Conference Proceedings 965 (New York, 2007).
43. C. Tsallis, C. Anteneodo, L. Borland and R. Osorio, Nonextensive statistical mechanics and economics, Physica A 324, 89 (2003).
44. M. Bologna, C. Tsallis and P. Grigolini, Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions, Phys. Rev. E 62, 2213 (2000).
45. F.D. Nobre and C. Tsallis, Infinite-range Ising ferromagnet - Thermodynamic limit within generalized statistical mechanics, Physica A 213, 337 (1995).
46. C. Anteneodo and C. Tsallis, Multiplicative noise: A mechanism leading to nonextensive statistical mechanics, J. Math. Phys. 44, 5194 (2003).
47. F.A. Tamarit, S.A. Cannas and C. Tsallis, Sensitivity to initial conditions in the Bak-Sneppen model of biological evolution, Eur. Phys. J. B 1, 545 (1998).
48. C. Anteneodo and C. Tsallis, Two-dimensional turbulence in pure-electron plasma: A nonextensive thermostatistical description, J. Mol. Liq. 71, 255 (1997).
49. L. Nivanen, A. Le Mehaute and Q.A. Wang, Generalized algebra within a nonextensive statistics, Rep. Math. Phys. 52, 437 (2003).
50. E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A 340, 95 (2004).
51. A.R. Plastino and A. Plastino, Non-extensive statistical mechanics and generalized Fokker-Planck equation, Physica A 222, 347 (1995).
52. C. Tsallis and D.J. Bukman, Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E 54, R2197 (1996).
53. M.A. Fuentes and M.O. Caceres, Computing the non-linear anomalous diffusion equation from first principles, Phys. Lett. A 372, 1236 (2008).
54. L. Borland, Ito-Langevin equations within generalized thermostatistics, Phys. Lett. A 245, 67 (1998).
55. A.M. Mariz and C. Tsallis, (2009) in progress.
56. W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Comm. Math. Phys. 56, 125 (1977). The reader should be aware that the domain of applicability of this theorem might, in spite of its declared hypothesis, not cover the case of Hamiltonians including long-range interactions at collective states such as the so-called QSS (quasi-stationary state). The 20 March 2006, in a telephone conversation with Prof. Werner Braun (while he was in Texas, and myself at the Santa Fe Institute, New Mexico), he shared with me [nearly literal transcription]: "We were thinking of a potential that has no singularities at the origin and elsewhere, and that decays sufficiently quickly at infinity... In this case we are sure that we are allowed to do all the derivatives and integrals that we do in the paper... Although I cannot be sure after so many years, my guess would be that our arguments are NOT valid if ( < d. In such a case, I would guess that strange structures could be present in phase space... I am not sure about all this, but this would be my guess. This conversation followed after the e-mail that, in 4 December 2005, I sent to both Prof. Braun and Prof. Hepp. The content of that e-mail (to which I received no reply) was: Dear colleagues, May I ask you a question about your theorem concerning the Vlasov equation? It plays some important role in one of the paradigmatic systems addressed by nonextensive statistical mechanics (see, for instance, http://tsallis.cat.cbpf.br/biblio.htm and also the attachment of the next e-mail, to be published momentarily by the European Physical Society). It is clear for me that the hypothesis assumed in that theorem are fulfilled for Hamiltonian systems that tend to occupy ALL of the associated Gibbs ( phase space. It is intuitively less clear for me that they are fulfilled if the dynamical occupancy in phase space is hierarchical, multifractal-like, possibly with ZERO LEBESGUE MEASURE (like the trajectories of the flights of an airline company, based in say Paris, like Air France, or in Houston, like Continental Airlines, etc). In such anomalous case, there are several numerical indications in the literature indicating the enormous influence that initial conditions play. I would certainly be pleased to understand better how these facts can be reconciled. Your clarifying comments would be greatly appreciated. Thank you very much. Best regards, Constantino Tsallis (Santa Fe Institute, New Mexico).
57. D. Broadhurst (1999), http://pi.lacim.uqam.ca/piDATA/feigenbaum.txt
58. F. Caruso and C. Tsallis, Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics, Phys. Rev. E 78, 021102 (2008).
59. q extensive, Proc. Natl. Acad. Sc. USA 102, 15377 (2005).
60. A. Rodriguez, V. Schwammle and C. Tsallis, Strictly and asymptotically scale-invariant probabilistic models of N correlated binary random variables having q-Gaussians as N → ∞ limiting distributions, JSTAT P09006 (2008).
61. R. Hanel, S. Thurner and C. Tsallis, Scale-invariant correlated probabilistic model yields q-Gaussians in the thermodynamic limit, preprint (2008).
62. E. Lutz, Anomalous diffusion and Tsallis statistics in an optical lattice, Phys. Rev. A 67, 051402(R) (2003).
63. M.O. Caceres, Computing a non-Maxwellian velocity distribution from first principles, Phys. Rev. E 67, 016102 (2003).
64. G.F. Mazenko, Vortex velocities in the O(n) symmetric time-dependent Ginzburg-Landau model, Phys. Rev. Lett. 78, 401 (1997).
65. H. Qian and G.F. Mazenko, Vortex dynamics in a coarsening two-dimensional XY model, Phys. Rev. E 68, 021109 (2003).
66. R. Albert and A.-L. Barabasi, Topology of evolving networks: Local events and universality, Phys. Rev. Lett. 85, 5234 (2000).
67. B.M. Boghosian, P.J. Love, P.V. Coveney, I.V. Karlin, S. Succi and J. Yepez, Galilean-invariant lattice-Boltzmann models with H-theorem, Phys. Rev. E 68, 025103(R) (2003).
68. B.M. Boghosian, P.J. Love, J. Yepez and P.V. Coveney, Galilean-invariant multi-speed entropic lattice Boltzmann models, in Anomalous Distributions, Nonlinear Dynamics and Nonextensivity, eds. H.L. Swinney and C. Tsallis, Physica D 193, 169 (2004).
69. B.M. Boghosian, General theory of Galilean-invariant entropic lattice Boltzmann models, in Complexity, Metastability and Nonextensivity, eds. C. Beck, G. Benedek, A. Rapisarda and C. Tsallis (World Scientific, Singapore, 2005), page 185.
70. A.M. Scarfone, P. Quarati, G. Mezzorani and M. Lissia, Analytical predictions of non-Gaussian distribution parameters for stellar plasmas, Astrophysics and Space Science 315, 353 (2008).
71. A. Rossani and A.M. Scarfone, Generalized Pearson distributions for charged particles interacting with an electric and/or a magnetic field, Physica A 388, 2354 (2009).
72. E. Mayoral and A. Robledo, Tsallis' q index and Mori's q phase transitions at edge of chaos, Phys. Rev. E 72, 026209 (2005).
73. H. Hernandez-Saldana and A. Robledo, Dynamics at the quasiperiodic onset of chaos, Tsallis q-statistics and Mori's q-phase transitions, Physica A 370, 286 (2006).
74. R.S. Mendes and C. Tsallis, Renormalization group approach to nonextensive statistical mechanics, Phys. Lett. A 285, 273 (2001).
75. S. Umarov, C. Tsallis, M. Gell-Mann and S. Steinberg, Symmetric (q,α)-stable distributions. Part I: First representation, cond-mat/0606038v2 (2008).
76. S. Umarov, C. Tsallis, M. Gell-Mann and S. Steinberg, Symmetric (q,α)-stable distributions. Part I: Second representation, cond-mat/0606040v2 (2008).
77. K.P. Nelson and S. Umarov, The relationship between Tsallis statistics, the Fourier transform, and nonlinear coupling, preprint (2008), 0811.3777 [cs.IT].
78. C. Tsallis, Dynamical scenario for nonextensive statitical mechanics, Physica A 340, 1 (2004).
79. L.F. Burlaga and A.F. Vinas, Triangle for the entropic index q of non-extensive statistical mechanic observed by Voyager 1 in the distant heliosphere, Physica A 356, 375 (2005).
80. N.O. Baella, private communication (2008).
81. P. Grassberger, Temporal scaling at Feigenbaum points and nonextensive thermodynamics, Phys. Rev. Lett. 95, 140601 (2005).
82. C. Tsallis, Comment on "Temporal scaling at Feigenbaum points and nonextensive thermodynamics" by P. Grassberger, cond-mat/0511213 (2005).
83. A. Robledo, Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points, Physica A 370, 449 (2006).
84. A. Robledo and L.G. Moyano, q-deformed statistical-mechanical property in the dynamics of trajectories en route to the Feigenbaum attractor, Phys. Rev. E 77, 032613 (2008).
85. U. Tirnakli, C. Beck and C. Tsallis, Central limit behavior of deterministic dynamical systems, Phys. Rev. E 75, 040106(R) (2007).
86. U. Tirnakli, C. Tsallis and C. Beck, A closer look at time averages of the logistic map at the edge of chaos, Phys. Rev. E 79, 056209 (2009).
87. This paper replies also to criticism by P. Grassberger, Proposed central limit behavior in deterministic dynamical systems, Phys. Rev. E 79, 057201 (2009).
88. C. Tsallis, Nonextensive statistical mechanics, anomalous diffusion and central limit theorems, Milan Journal of Mathematics 73, 145 (2005).
89. J.P Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).
90. C. Tsallis and S.M.D. Queiros, Nonextensive statistical mechanics and central limit theorems I - Convolution of independent random variables and q-product, in Complexity, Metastability and Nonextensivity, eds. S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda and C. Tsallis, American Institute of Physics Conference Proceedings 965, 8 (New York, 2007).
91. S.M.D. Queiros and C. Tsallis, Nonextensive statistical mechanics and central limit theorems II - Convolution of q-independent random variables, in Complexity, Metastability and Nonextensivity, eds. S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda and C. Tsallis, American Institute of Physics Conference Proceedings 965, 21 (New York, 2007).
92. H.J. Hilhorst (2008), private communication.
93. H.J. Hilhorst, Braz. J. Phys. (2009), in the present volume.
94. C. Tsallis and E.M.F. Curado, unpublished.
95. C. Vignat and A. Plastino, Central limit theorem and deformed exponentials, J. Phys. A 40, F969 (2007).
96. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999).
97. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000).
98. R. Hilfer, ed., Applications Of Fractional Calculus In Physics (World Scientific, Singapore, 2000).
99. P. Douglas, S. Bergamini and F. Renzoni, Tunable Tsallis distributions in dissipative optical lattices, Phys. Rev. Lett. 96, 110601 (2006).
100. B. Liu and J. Goree, Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma, Phys. Rev. Lett. 100, 055003 (2008).
101. B. Liu, J. Goree and Y. Feng, Non-Gaussian statistics and superdiffusion in a driven-dissipative dusty plasma, Phys. Rev. E 78, 046403 (2008).
102. R.G. DeVoe, Power-law distributions for a trapped ion interacting with a classical buffer gas, Phys. Rev. Lett. 102, 063001 (2009).
103. A. Upadhyaya, J.-P. Rieu, J.A. Glazier and Y. Sawada, Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates, Physica A 293, 549 (2001).
104. K.E. Daniels, C. Beck and E. Bodenschatz, Defect turbulence and generalized statistical mechanics, Physica D 193, 208 (2004).
105. L.F. Burlaga, N.F. Ness and M.H. Acuna, Multiscale structure of magnetic fields in the heliosheath, J. Geophysical Res.-Space Physics 111, A09112 (2006).
106. L.F. Burlaga, A.F. Vinas, N.F. Ness and M.H. Acuna, Tsallis statistics of the magnetic field in the heliosheath, Astrophys. J. 644, L83 (2006).
107. L.F. Burlaga, A. F-Vinas and C. Wang, Tsallis distributions of magnetic field strength variations in the heliosphere: 5 to 90 AU, J. Geophysical Res.-Space Physics 112, A07206 (2007).
108. L.F. Burlaga, N.F. Ness, M.H. Acuna, Magnetic fields in the heliosheath and distant heliosphere: Voyager 1 and 2 observations during 2005 and 2006, Astrophys. J. 668, 1246 (2007).
109. L.F. Burlaga, N.F. Ness and M.H. Acuna, Magnetic field strength fluctuations and temperature in the heliosheath, Astrophys. J. 691, L82 (2009).
110. R. Arevalo, A. Garcimartin and D. Maza, Anomalous diffusion in silo drainage, Eur. Phys. J. E 23, 191 (2007).
111. R. Arevalo, A. Garcimartin and D. Maza, A non-standard statistical approach to 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 143 (2007).
112. A. Pluchino, A. Rapisarda and C. Tsallis, Nonergodicity and central limit behavior in long-range Hamiltonians, Europhys. Lett. 80, 26002 (2007).
113. A. Pluchino and A. Rapisarda, Nonergodicity and central limit behavior for systems with long-range interactions, SPIE 2, 6802-32 (2007).
114. 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 the HMF model, Physica A 387, 3121 (2008).
115. For criticism and its rebuttal see A. Figueiredo, T.M. Rocha Filho and M.A. Amato, Ergodicity and central limit theorem in systems with long-range interactions, Europhys. Lett. 83, 30011 (2008)
116. A. Pluchino, A. Rapisarda and C. Tsallis, Comment on "Ergodicity and central limit theorem in systems with long-range interactions" by Figueiredo A. et al, Europhys. Lett. 85, 60006 (2009)
117. A. Figueiredo, T.M. Rocha Filho and M.A. Amato, Reply to to the Comment by A. Pluchino et al, Europhys. Lett. 85, 60007 (2009).
118. T. Dauxois, S. Ruffo, E. Arimondo and M. Wilkens, eds., Dynamics and Thermodynamics of Systems with Long Range Interactions, Lecture Notes in Physics 602 (Springer, Berlin, 2002).
119. A. Antoniazzi, D. Fanelli, J. Barre, P.-H. Chavanis, T. Dauxois and S. Ruffo, Maximum entropy principle explains quasistationary states in systems with long-range interactions: The example of the Hamiltonian mean-field model, Phys. Rev. E 75, 011112 (2007).
120. G. Miritello, A. Pluchino and A. Rapisarda, Central limit violation in the Kuramoto model at the 'edge of chaos', preprint (2009), 0903.4400 [cond-mat.stat-mech].
121. G. Ruiz and C. Tsallis, Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps, Eur. Phys. J. B 67, 577 (2009).
122. F. Caruso, A. Pluchino, V. Latora, S. Vinciguerra and A. Rapisarda, Analysis of self-organized criticality in the Olami-Feder-Christensen model and in real earthquakes, Phys. Rev. E 75, 055101(R)(2007).
123. B. Bakar and U. Tirnakli, Analysis of self-organized criticality in Ehrenfest's dog-flea model, Phys. Rev. E 79, 040103 (2009).
124. M. Suvakov and B. Tadic, Collective charge fluctuations in single-electron processes on nanonetworks, J. Stat. Mech. P02015 (2009).
125. L.Borland, Option pricing formulas based on a non-Gaussian stock price model, Phys. Rev. Lett. 89, 098701 (2002).
126. L. Borland, A theory of non-gaussian option pricing, Quantitative Finance 2, 415 (2002).
127. C. Tsallis, C. Anteneodo, L. Borland and R. Osorio, Nonextensive statistical mechanics and economics, Physica A 324, 89 (2003).
128. 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, 2004).
129. L. Borland and J.-P. Bouchaud, A non-Gaussian option pricing model with skew, Quant. Finance 4, 499 (2004).
130. L. Borland, The pricing of stock options, in Nonextensive Entropy - Interdisciplinary Applications, eds. M. Gell-Mann and C. Tsallis (Oxford University Press, New York, 2004).
131. L. Borland, Long-range memory and nonextensivity in financial markets, in Nonextensive Statistical Mechanics: New Trends, New perspectives, eds. J.P. Boon and C. Tsallis, Europhysics News 36, 228 (2005).
132. L. Borland, A non-Gaussian model of stock returns: option smiles, credit skews, and a multi-time scale memory, in Noise and Fluctuations in Econophysics and Finance, eds. D. Abbott, J.-P. Bouchaud, X. Gabaix and J.L. McCauley, Proc. of SPIE 5848, 55 (SPIE, Bellingham, WA, 2005).
133. S.M.D. Queiros, On non-Gaussianity and dependence in financial in time series: A nonextensive approach, Quant. Finance 5, 475 (2005)
134. L. Borland, A non-Gaussian stock price model: Options, credit and a multi-timescale memory, in Complexity and Nonextensivity: New Trends in Statistical Mechanics, eds. M. Sakagami, N. Suzuki and S. Abe, Prog. Theor. Phys. Suppl. 162, 155 (2006).
135. A.A.G. Cortines and R. Riera Freire, Non-extensive behavior of a stock market index at microscopic time scales, Physica A 377, 181 (2007).
136. T.D. Frank, Nonextensive cutoff distributions of postural sway for the old and the young, Physica A 388, 2503 (2009).
137. R.M. Pickup, R. Cywinski, C. Pappas, B. Farago and P. Fouquet, Generalized spin glass relaxation, Phys. Rev. Lett. 102, 097202 (2009).