Nonextensive statistical mechanics: A brief introduction (original) (raw)

References

  1. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479 (1988)
    MathSciNet MATH Google Scholar
  2. Curado, E.M.F., Tsallis, C.: Generalized statistical mechanics: connection with thermodynamics. J. Phys. A 24, L69 (1991) [Corrigenda: 24, 3187 (1991) and 25, 1019 (1992)]
  3. Tsallis, C., Mendes, R.S., Plastino, A.R.: The role of constraints within generalized nonextensive statistics. Physica A 261, 534 (1998)
    Article Google Scholar
  4. Beck, C., Schlogl, F.: Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1993)
  5. Abe, S., Rajagopal, A.K.: Microcanonical foundation for systems with power-law distributions. J. Phys. A 33, 8733 (2000)
    MATH Google Scholar
  6. Abe, S., Martínez, S., Pennini, F., Plastino, A.: Nonextensive thermodynamic relations. Phys. Lett. A 281, 126 (2001)
    MATH Google Scholar
  7. Salinas, S.R.A., Tsallis, C. (eds.): Nonextensive Statistical Mechanics and Thermodynamics, Brazilian Journal of Physics 29 (1999); Abe, S., Okamoto, Y. (eds.): Nonextensive Statistical Mechanics and Its Applications, Series Lecture Notes in Physics (Springer-Verlag, Berlin, 2001); Kaniadakis, G., Lissia, M., Rapisarda, A. (eds.): Non Extensive Thermodynamics and Physical Applications, Physica A 305 (Elsevier, Amsterdam, 2002); Grigolini, P., Tsallis, C., West, B.J. (eds.): Classical and Quantum Complexity and Nonextensive Thermodynamics, Chaos, Solitons and Fractals 13, Number 3 (Pergamon-Elsevier, Amsterdam, 2002); Gell-Mann, M., Tsallis, C. (eds.): Nonextensive Entropy - Interdisciplinary Applications, (Oxford University Press, New York, 2004), in press; Swinney, H.L., Tsallis, C.: Anomalous Distributions, Nonlinear Dynamics, and Nonextensivity, Physica D (Elsevier, Amsterdam, 2004), in press. A regularly updated bibliography on the subject can be found at http://tsallis.cat.cbpf.br/biblio.htm
  8. Krylov, N.S.: Nature 153, 709 (1944); Krylov, N.S.: Works on the Foundations of Statistical Physics, translated by A.B. Migdal, Ya. G. Sinai and Yu. L. Zeeman, Princeton Series in Physics (Princeton University Press, Princeton, 1979)
  9. Cohen, E.G.D.: Statistics and dynamics. Physica A 305, 19 (2002)
    Article MathSciNet MATH Google Scholar
  10. Baranger, M.: Why Tsallis statistics? Physica A 305, 27 (2002)
  11. Tsallis, C., Plastino, A.R., Zheng, W.M.: Power-law sensitivity to initial conditions - New entropic representation. Chaos, Solitons and Fractals 8, 885 (1997)
    Google Scholar
  12. Baldovin, F., Robledo A.: Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions. Europhys. Lett. 60, 518 (2002); Phys. Rev. E 66, 8045104 (2002)
    Article MathSciNet Google Scholar
  13. Baldovin, F., Robledo, A.: Nonextensive Pesin identity. Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map. cond-mat/0304410 (2003)
  14. Lyra, M.L., Tsallis, C.: Nonextensivity and multifractality in low-dimensional dissipative systems. Phys. Rev. Lett. 80, 53 (1998)
    Article Google Scholar
  15. Latora, V., Baranger, M., Rapisarda, A., Tsallis, C.: The rate of entropy increase at the edge of chaos. Phys. Lett. A 273, 97 (2000)
    Article MathSciNet Google Scholar
  16. de Moura, F.A.B.F., Tirnakli, U., Lyra, M.L.: Convergence to the critical attractor of dissipative maps: Log-periodic oscillations, fractality and nonextensivity. Phys. Rev. E 62, 6361 (2000)
    Article Google Scholar
  17. Borges, E.P., Tsallis, C., Ananos, G.F.J., Oliveira, P.M.C.: Nonequilibrium probabilistic dynamics at the logistic map edge of chaos. Phys. Rev. Lett. 89, 254103 (2002)
    Article Google Scholar
  18. Weinstein, Y.S., Lloyd, S., Tsallis, C.: Border between between regular and chaotic quantum dynamics. Phys. Rev. Lett. 89, 214101 (2002); Weinstein, Y.S, Tsallis, C., Lloyd, S.: On the emergence of nonextensivity at the edge of quantum chaos, in: Elze, H.T. (ed.): Decoherence and Entropy in Complex Systems, Lecture Notes in Physics (Springer, Heidelberg, 2003), in press
    Article Google Scholar
  19. Tirnakli, U., Tsallis, C., Lyra, M.L.: Asymmetric unimodal maps at the edge of chaos. Phys. Rev. E 65, 036207 (2002)
    Article Google Scholar
  20. Tirnakli, U.: Two-dimensional maps at the edge of chaos: Numerical results for the Henon map. Phys. Rev. E 66, 066212 (2002); Borges, E.P., Tirnakli, U.: Mixing and relaxation dynamics of the Henon map at the edge of chaos, Physica D (2004), in press [cond-mat/0302616]
    Article Google Scholar
  21. Baldovin, F., Brigatti, E., Tsallis, C.: Quasi-stationary states in low dimensional Hamiltonian systems. Phys. Lett. A 320, 254 (2004)
    Article Google Scholar
  22. Ananos, G.F.J., Baldovin, F., Tsallis, C.: Nonstandard sensitivity to initial conditions and entropy production in standard maps. Preprint (2003)
  23. Tsekouras, G.A., Provata, A., Tsallis, C.: Nonextensivity of the cyclic lattice Lotka Volterra model. Phys. Rev. E (1 Jan 2004), in press
  24. Boghosian, B.M., Love, P.J., Coveney, P.V., Karlin, I.V., Succi S., Yepez, J.: Galilean-invariant lattice-Boltzmann models with _H_-theorem. Phys. Rev. E 68, 025103(R) (2003); Boghosian, B.M., Love, P., Yepez, J., Coveney, P.V.: Galilean-invariant multi-speed entropic lattice Boltzmann models. Physica D (2004), in press
    Article Google Scholar
  25. Albert, R., Barabasi, A.L.: Phys. Rev. Lett. 85, 5234 (2000)
    Article Google Scholar
  26. Antoni, M., Ruffo, S.: Phys. Rev. E 52, 2361 (1995)
    Article Google Scholar
  27. Anteneodo, C., Tsallis, C.: Breakdown of the exponential sensitivity to the initial conditions: Role of the range of the interaction. Phys. Rev. Lett. 80, 5313 (1998)
    Article Google Scholar
  28. Campa, A., Giansanti, A., Moroni, D., Tsallis, C.: Classical spin systems with long-range interactions: Universal reduction of mixing. Phys. Lett. A 286, 251 (2001)
    Article MATH Google Scholar
  29. Latora, V., Rapisarda, A., Tsallis, C.: Non-Gaussian equilibrium in a long-range Hamiltonian system. Phys. Rev. E 64, 056134 (2001)
    Article Google Scholar
  30. Montemurro, M.A., Tamarit, F., C. Anteneodo: Aging in an infinite-range Hamiltonian system of coupled rotators. Phys. Rev. E 67, 031106 (2003)
    Article Google Scholar
  31. Cabral, B.J.C., Tsallis, C.: Metastability and weak mixing in classical long-range many-rotator system. Phys. Rev. E 66, 065101(R) (2002)
    Article Google Scholar
  32. Tsallis, C.: Comment on “A critique of _q_-entropy for thermal statistics” by M. Nauenberg. Phys. Rev. E (2004), in press
  33. Moyano, L.G., Baldovin, F., Tsallis, C.: Zeroth principle of thermodynamics in aging quasistationary states. cond-mat/0305091
  34. Nobre, F.D., Tsallis, C.: Classical infinite-range-interaction Heisenberg ferromagnetic model: Metastability and sensitivity to initial conditions. Phys. Rev. E 68, 036115 (2003)
    Article Google Scholar
  35. Borges, E.P., Tsallis, C., Giansanti, A., Moroni, D.: Dinamica de um sistema nao extensivo de rotores classicos anisotropicos acoplados, in Tendencias da Fisica Estatistica no Brasil, ed. T. Tome, volume honoring S.R.A. Salinas (Editora Livraria da Fisica, Sao Paulo, 2003), page 84
  36. Doye, J.P.K.: Network topology of a potential energy landscape: A static scale-free network. Phys. Rev. Lett. 88, 238701 (2002)
    Article Google Scholar
  37. Lutz, E.: Anomalous diffusion and Tsallis statistics in an optical lattice. Phys. Rev. A 67, 051402(R) (2003)
    Google Scholar
  38. Plastino, A.R. and Plastino, A.: Non-extensive statistical mechanics and generalized Fokker-Planck equation. Physica A 222, 347 (1995); Tsallis, C., Bukman, D.J.: Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis. Phys. Rev. E 54, R2197 (1996); Curado, E.M.F., Nobre, F.D.: Derivation of nonlinear Fokker-Planck equations by means of approximations to the master equation. Phys. Rev. E 67, 021107 (2003)
    Article MathSciNet Google Scholar
  39. Alemany, P.A. and Zanette, D.H.: Fractal random walks from a variational formalism for Tsallis entropies. Phys. Rev. E 49, R956 (1994); Tsallis, C., Levy, S.V.F, de Souza A.M.C., Maynard, R.: Statistical-mechanical foundation of the ubiquity of Levy distributions in nature. Phys. Rev. Lett. 75, 3589 (1995) [Erratum: 77, 5442 (1996)]; Prato, D., Tsallis, C.: Nonextensive foundation of Levy distributions. Phys. Rev. E 60, 2398 (1999)
    Google Scholar
  40. Bologna, M., Tsallis, C., Grigolini, P.: Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions. Phys. Rev. E 62, 2213 (2000)
    Article Google Scholar
  41. Tsallis, C., Lenzi, E.K.: in Hilfer, R., et al. (eds.) Strange Kinetics. Chem. Phys. 284, 341 (2002) [Erratum: 287, 341 (2002)]; Lenzi, E.K., Mendes, R.S., Tsallis, C.: Crossover in diffusion equation: Anomalous and normal behaviors. Phys. Rev. E 67, 031104 (2003)
    Google Scholar
  42. Anteneodo, C., Tsallis, C.: Multiplicative noise: A mechanism leading to nonextensive statistical mechanics. J. Math. Phys. 44, 5194 (2003)
    Article Google Scholar
  43. Caceres, M.O.: Phys. Rev. E 67, 016102 (2003)
    Article Google Scholar
  44. Tsallis, C., Bemski, G., Mendes, R.S.: Is re-association in folded proteins a case of nonextensivity? Phys. Lett. A 257, 93 (1999)
    Article Google Scholar
  45. Tsallis, C., Anjos, J.C., Borges, E.P.: Fluxes of cosmic rays: A delicately balanced stationary state. Phys. Lett. A 310, 372 (2003)
    Article Google Scholar
  46. Beck, C.: Foundations of nonextensive statistical mechanics. Phys. Rev. Lett. 87, 180601 (2001)
    Article Google Scholar
  47. Beck, C., Lewis, G.S., Swinney, H.L.: Measuring non-extensivity parameters in a turbulent Couette-Taylor flow. Phys. Rev. E 63, 035303 (2001)
    Article Google Scholar
  48. Arimitsu, N., Arimitsu, T.: Analysis of velocity derivatives in turbulence based on generalized statistics. Europhys. Lett. 60, 60 (2002)
    Article Google Scholar
  49. Borland, L.: Closed form option pricing formulas based on a non-Gaussian stock price model with statistical feedback. Phys. Rev. Lett. 89, 098701 (2002); Tsallis, C., Anteneodo, C., L. Borland and R. Osorio: Nonextensive statistical mechanics and economics. Physica A 324, 89 (2003)
    Article Google Scholar
  50. Bediaga, I., Curado, E.M.F. and Miranda, J.: A nonextensive thermodynamical equilibrium approach in \(e^+e^- \rightarrow hadrons\). Physica A 286, 156 (2000)
    Article Google Scholar
  51. Upadhyaya, A., Rieu, J.-P., Glazier, J.A., Sawada, Y.: Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates. Physica A 293, 549 (2001)
    Article Google Scholar
  52. Rosso, O.A., Martin, M.T., Plastino, A.: Brain electrical activity analysis using wavelet based informational tools. Physica A 313, 587 (2002)
    Article MATH Google Scholar
  53. Montemurro, M.A.: Beyond the Zipf-Mandelbrot law in quantitative linguistics. Physica A 300, 567 (2001)
    Article MATH Google Scholar
  54. Tsallis, C., Borges, E.P.: Nonextensive statistical mechanics - Applications to nuclear and high energy physics, in: Antoniou, N.G., Diakonos, F.K., Ktorides, C.N. (eds.) Proc. 10th International Workshop on Multiparticle Production - Correlations and Fluctuations in QCD (World Scientific, Singapore, 2003), p. 326
  55. Tsallis, C., Prato, D., Plastino, A.R.: Nonextensive statistical mechanics: Some links with astronomical phenomena, to appear in the Proceedings of the XIth United Nations / European Space Agency Workshop on Basic Space Sciences, Office for Outer Space Affairs / United Nations (Cordoba, 9-13 September 2002), ed. H. Haubold, special issue of Astrophysics and Space Science (Kluwer Academic Publishers, Dordrecht, 2003)
  56. Borges, E.P.: Comment on “The individual success of musicians, like that of physicists, follows a streched exponential distribution’”. Eur. Phys. J. B 30, 593 (2002)
    Google Scholar
  57. Malacarne, L.C., R.S. Mendes, Lenzi, E.K.: _q_-Exponential distribution in urban agglomeration. Phys. Rev. E 65, 017106 (2002)
    Article Google Scholar
  58. Abe, S., Suzuki, N.: Itineration of the Internet over nonequlibrium stationary states in Tsallis statistics. Phys. Rev. E 67, 016106 (2003)
    Article Google Scholar
  59. Tsallis, C. Stariolo, D.A.: Generalized simulated annealing. Physica A 233, 395 (1996); a preliminary version appeared (in English) as Notas de Fisica/CBPF 026 (June 1994); Penna, T.J.P.: Traveling salesman problem and Tsallis statistics. Phys. Rev. E 51, R1 (1995); Mundim, K.C., Tsallis, C.: Geometry optimization and conformational analysis through generalized simulated annealing. Int. J. Quantum Chem. 58, 373 (1996); Andricioaei, I., Straub, J.E.: Phys. Rev. E 53, R3055 (1996); Moret, M.A., Pascutti, P.G., Bisch, P.M., Mundim, K.C.: Generalized simulated annealing algorithms using Tsallis statistics: Application to conformational optimization of a tetrapeptide. J. Comp. Chemistry 19, 647 (1998); Guo, L., Ellis, D.E., Mundim, K.C.: Macrocycle-macrocycle interactions within one-dimensional Cu phthalocyanine chains. Journal of Porphyrins and Phthalocyanines 3, 196 (1999); Moret, M.A., Bisch, P.M., Vieira, F.M.C.: Algorithm for multiple minima search Phys. Rev. E 57, R2535 (1998); Berner, A., Fuks, D., Ellis, D.E., Mundim, K.C. and Dorfman, S.: Formation of nano-crystalline structure at the interface in Cu-C composite. Applied Surface Science 144-145, 677 (1999); Serra, P., Stanton, A.F., Kais, S., Bleil, R.E.: Comparison study of pivot methods for global optimization. J. Chem. Phys. 106, 7170 (1997); Xiang, Y., Sun, D.Y., Fan, W., Gong, X.G.: Generalized simulated annealing algorithm and its aplication to the Thomson model. Phys. Lett. A 233, 216 (1997); Andricioaei, I., Straub, J.E.: On Monte Carlo and molecular dynamics inspired by Tsallis statistics: Methodology, optimization and applications to atomic clusters. J. Chem. Phys. 107, 9117 (1997); U.H.E. Hansmann , M. Masuya and Y. Okamoto: Characteristic temperatures of folding of a small peptide. Proc. Natl. Acad. Sci. USA 94, 10652 (1997); M.R. Lemes, C.R. Zacharias and A. Dal Pino Jr.: Generalized simulated annealing: Application to silicon clusters. Phys. Rev. B 56, 9279 (1997); Yu, Z.X., Mo, D.: Generalized simulated annealing algorithm applied in the ellipsometric inversion problem. Thin Solid Films 425, 108 (2003)
    Article Google Scholar
  60. Lesche, B.: Instabilities of Renyi entropies. J. Stat. Phys. 27, 419 (1982)
    MathSciNet Google Scholar
  61. Abe, S.: Stability of Tsallis entropy and instabilities of Renyi and normalized Tsallis entropies: A basis for _q_-exponential distributions. Phys. Rev. E 66, 046134 (2002)
    Article Google Scholar
  62. Abe, S.: Tsallis entropy: How unique? to appear in Continuum Mechanics and Thermodynamics (2003) [cond-mat/0305087]
  63. Landsberg, P.T., Vedral, V.: Distributions and channel capacities in generalized statistical mechanics. Phys. Lett. A 247, 211 (1998); Rajagopal, A.K., Abe, S.: Implications of form invariance to the structure of nonextensive entropies. Phys. Rev. Lett. 83, 1711 (1999)
    MathSciNet MATH Google Scholar
  64. Luzzi, R., Vasconcellos, A.R, Galvao Ramos, J.: Trying to make sense of disorder Science. 298, 1171 (2002)
  65. Abe, S., Rajagopal, A.K.: Revisiting disorder and Tsallis statistics. Science 300, 249 (2003)
    Article Google Scholar
  66. Plastino, A.: Revisiting disorder and Tsallis statistics. Science 300, 250 (2003)
    Google Scholar
  67. Latora, V., Rapisarda, A., Robledo, A.: Revisiting disorder and Tsallis statistics. Science 300, 250 (2003)
    Google Scholar
  68. Nauenberg, M.: A critique of _q_-entropy for thermal statistics. Phys. Rev. E 67, 036114 (2003)
    Article Google Scholar
  69. Zanette, D.H., Montemurro, M.A.: Thermal measurements of stationary nonequilibrium systems: A test for generalized thermostatistics. Phys. Lett. A 316, 184 (2003)
    Article MATH Google Scholar

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