Nonextensive statistical mechanics, anomalous diffusion and central limit theorems

Milan Journal of Mathematics

Volumn 73, Issue 1, 2005, Pages 145-176

  Tsallis, Constantino ^a,b  

^a SANTA FE INSTITUTE   (United States)

^b CENTRO BRASILEIRO DE PESQUISAS FÍSICAS   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 27844494292     PISSN: 14249286     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00032-005-0041-1     Document Type: Review

References (59)

1. L. Boltzmann, Weitere Studien u̇ber das Wȧrmegleichgewicht unter Gas moleku̇len. [Further Studies on Thermal Equilibrium Between Gas Molecules], Wien, Ber. 66 (1872), 275.
2. J.W. Gibbs, Elementary Principles in Statistical Mechanics - Developed with Especial Reference to the Rational Foundation of Thermodynamics. C. Scribner's Sons, New York, 1902; Yale University Press, New Haven, 1948; OX Bow Press, Woodbridge, Connecticut, 1981. In page 35: "In treating of the canonical distribution, we shall always suppose the multiple integral in equation (4.41) [the partition function, as we call it nowadays] to have a finite value, as otherwise the coefficient of probability vanishes, and the law of distribution becomes illusory. This will exclude certain cases, but not such apparently, as will affect the value of our results with respect to their bearing on thermodynamics. It will exclude, for instance, cases in which the system or parts of it can be distributed in unlimited space [...]. It also excludes many cases in which the energy can decrease without limit, as when the system contains material points which attract one another inversely as the squares of their distances. [...]. For the purposes of a general discussion, it is sufficient to call attention to the assumption implicitly involved in the formula (4.41)."
3. A. Einstein, Theorie der Opaleszenz von homogenen Flu̇ssigkeiten und Flu̇ssigkeitsgemischen in der Nȧhe des kritischen Zustandes, Annalen der Physik 33, 1275 (1910). ["Usually W is put equal to the number of complex-ions...In order to calculate W, one needs a complete (molecular-mechanical) theory of the system under consideration. Therefore it is dubious whether the Boltzmann principle has any meaning without a complete molecular-mechanical theory or some other theory which describes the elementary processes. S = R/N log W + const. seems without content, from a phenomeno-logical point of view, without giving in addition such an Elementartheorie." (Translation: Abraham Pais, Subtle is the Lord..., Oxford University Press, 1982)].
4. M. Gell-Mann and C. Tsallis, eds., Nonextensive Entropy - Interdisciplinary Applications (Oxford University Press, New York, 2004);
5. H.L. Swinney and C. Tsallis, eds., Anomalous Distributions, Nonlinear Dynamics and Nonextensivity, Physica D 193 (2004).
6. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988); for full bibliography see http://tsallis.cat.cbpf. br/biblio.htm.
7. E.M.F. Curado and C. Tsallis, Generalized statistical mechanics: connection with thermodynamics, J. Phys. A 24, L69 (1991); Corrigenda: 24, 3187 (1991) and 25, 1019 (1992).
8. C. Tsallis, R.S. Mendes and A.R. Plastino, The role of constraints within generalized nonextensive statistics, Physica A 261, 534 (1998).
9. i is called the Gibbs distribution. We shall give no justification for this dogma; even a physicist like Ruelle disposes of this question as "deep and incompletely clarified"."
10. C. Tsallis, What are the numbers that experiments provide?, Quimica Nova 17, 468 (1994).
11. D. Prato and C. Tsallis, Nonextensive foundation of Lévy distributions, Phys. Rev. E 60, 2398 (1999).
12. A.M.C. de Souza and C. Tsallis, Student's t- and r-distributions: Unified derivation from an entropic variational principle, Physica A 236, 52 (1997).
13. A. Einstein, Uber die von der molekular kinetischen Theorie der Warme gefordete Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen, Ànnalen der Physik 17, 549 (1905).
14. A.R. Plastino and A. Plastino, Non-extensive statistical mechanics and generalized Fokker-Planck equation, Physica A 222, 347 (1995)1.
15. 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).
16. E.M.F. Curado and F.D. Nobre, Derivation of nonlinear Fokker-Planck equations by means of approximations to the master equation, Phys. Rev. E 67, 021107 (2003);
17. F.D. Nobre, E.M.F. Curado and G. Rowlands, A procedure for obtaining general nonlinear Fokker-Planck equations, Physica A 334, 109 (2004).
18. L. Nivanen, A. Le Mehaute and Q.A. Wang, Generalized algebra within a nonextensive statistics, Rep. Math. Phys. 52, 437 (2003);
19. E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A 340, 95 (2004).
20. 1/(1-q).
21. q extensive or nonextensive?, in Complexity, Metastbility and Nonextensivity, eds. C. Beck, A. Rapisarda and C. Tsallis (World Scientific, Singapore, 2005), to appear [cond-mat/0409631];
22. q for systems with N ≤ 3 specially correlated subsystems, International Journal of Bifurcation and Chaos, to appear: Proceedings of the Summer School and Conference on Complexity in Science and Society, ed. T. Bountis (Fatras and Olympia, 14-26 July, 2004) [cond-mat/0411073].
23. q with the appropriate value of q is extensive, and is the one to be used. If the global correlation is of some different nature, even more complex than the one focused on in this paper, a still different entropic form might be necessary.
24. H. Suyari and M. Tsukada, Law of error in Tsallis statistics, IEEE Transaction on Information Theory 51, 753 (2005);
25. H. Suyari, q-Stirling's formula in Tsallis statistics, cond-mat/0401541;
26. H. Suyari, Mathematical structure derived from the q-multinomial coefficient in Tsallis statistics, cond-mat/0401546.
27. D.J. Watts and S.H. Strogatz, Collective dynamics of small-world networks, Nature 393, 440 (1998).
28. R. Albert and A.-L. Barabasi, Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47 (2002).
29. R. Albert and A.-L. Barabasi, Topology of evolving networks: Local events and universality, Phys. Rev. Lett. 85, 5234 (2000); the analytically exact distribution which is found in this paper is, although writen in a slightly different form, a q-exponential with q = 2m(2-r)+1-p-r/m(3-2r)+1-p-r, where (m, p, r) are parameters of the specific growth model.
30. J.P.K. Doye, Network topology of a potential energy landscape: A static scale-free network, Phys. Rev. Lett. 88, 238701 (2002);
31. J.P.K. Doye and C.P. Massen, Characterization of the network topology of the energy landscapes of atomic clusters, J. Chem. Phys. 122, 084105 (2005); the distributions that are presented in this paper can be shown to be numerically very close to a q-exponential with q sime; 2.
32. D.J.B. Soares, C. Tsallis, A.M. Mariz and L.R. da Silva, Preferential attachment growth model and nonextensive statistical mechanics. Europhys. Lett. 70, 70 (2005).
33. C. Tsallis, A.R. Plastino and W.-M. Zheng, Power-law sensitivity to initial conditions - New entropic representation, Chaos, Solitons and Fractals 8, 885 (1997).
34. F. Baldovin and A. Robledo, Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions, Europhys. Lett. 60, 518 (2002);
35. F. Baldovin and A. Robledo, Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics, Phys. Rev. E 66, R045104 (2002).
36. V. Latora and M. Baranger, Kolmogorov-Sinai entropy rate versus physical entropy, Phys. Rev. Lett. 82, 520 (1999).
37. V. Latora, M. Baranger, A. Rapisarda and C. Tsallis, The rate of entropy increase at the edge of chaos, Phys. Lett. A 273, 97 (2000).
38. F. Baldovin and A. Robledo, Nonextensive Pesin identity. Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map., Phys. Rev. E 69, 045202(R) (2004).
39. G.F.J. Ananos, F. Baldovin and C. Tsallis, Anomalous sensitivity to initial conditions and entropy production in standard maps: Nonextensive approach, Eur. Phys. J. B (2005), in press [cond-mat/0403656].
40. G. Casati, C. Tsallis and F. Baldovin, Linear instability and statistical laws of physics, cond-mat/0507504.
41. G. Casati and T. Prosen, Mixing property of triangular billiards, Phys. Rev. Lett. 83, 4729 (1999);
42. G. Casati and T. Prosen, Triangle map: A model of quantum chaos, Phys. Rev. Lett. 85, 4261 (2000).
43. G.A. Tsekouras, A. Provata and C. Tsallis, Nonextensivity of the cyclic lattice Lotka Volterra model, Phys. Rev. E 69, 016120 (2004).
44. C. Anteneodo, Entropy production in the cyclic lattice Lotka-Volterra model, Eur. Phys. J. B 42, 271 (2004).
45. 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).
46. C. Tsallis and A.M.C. Souza, Constructing a statistical mechanics for Beck-Cohen superstatistics, Phys. Rev. E 67, 026106 (2003);
47. see also A.M.C. Souza and C. Tsallis, Stability of the entropy for superstatistics, Phys. Lett. A 319, 273 (2003);
48. A.M.C. Souza and C. Tsallis, Stability analysis of the entropy for superstatistics, Physica A 342, 132 (2004).
49. C. Beck and E.G.D. Cohen, Superstatistics, Physica A 322, 267 (2003);
50. see also C. Beck, Superstatistics: Theory and Applications, in Nonadditive entropy and nonextensive statistical mechanics, ed. M. Sugiyama, Continuum Mechanics and Thermodynamics 16, 293 (Springer-Verlag, Heidelberg, 2004);
51. E.G.D. Cohen, Superstatistics, in Anomalous Distributions, Nonlinear Dynamics and Nonextensivity, eds. H.L. Swinney and C. Tsallis, Physica D 193, 35 (2004).
52. E.M.F. Curado, General aspects of the thermodynamical formalism, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S.R.A. Salinas and C. Tsallis, Braz. J. Phys. 29, 36 (1999);
53. E.M.F. Curado and F.D. Nobre, On the stability of analytic entropic forms, Physica A 335, 94 (2004).
54. C. Anteneodo and A.R. Plastino, Maximum entropy approach to stretched exponential probability distributions, J. Phys. A 32, 1089 (1999).
55. G. Kaniadakis, Nonlinear kinetics underlying generalized statistics, Physica A 296, 405 (2001);
56. G. Kaniadakis, Statistical mechanics in the context of special relativity, Phys. Rev. E 66, 0561125 (2002);
57. G. Kaniadakis, M. Lissia and A.M. Scarfone, Deformed logarithms and entropies, Physica A 340, 41 (2004).
58. M. Gell-Mann, The Quark and the Jaguar - Adventures in the Simple and the Complex, (W.H. Freeman, New York, 1994). Plectics is the denomination suggested by M. Gell-Mann for the emerging science of the so called complex systems.
59. E.G.D. Cohen, Boltzmann and Einstein: Statistics and dynamics - An unsolved problem, Boltzmann Award Lecture at Statphys-Bangalore-2004, Pramana 64, 635 (2005).