Skip to main content
SpringerLink
Log in

Nonextensive statistical mechanics: A brief introduction

  • Original article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract.

Boltzmann-Gibbs statistical mechanics is based on the entropy \(S_{\rm BG} = -k \sum_{i = 1}^W p_i \ln p_i\). It enables a successful thermal approach to ubiquitous systems, such as those involving short-range interactions, markovian processes, and, generally speaking, those systems whose dynamical occupancy of phase space tends to be ergodic. For systems whose microscopic dynamics is more complex, it is natural to expect that the dynamical occupancy of phase space will have a less trivial structure, for example a (multi)fractal or hierarchical geometry. The question naturally arises whether it is possible to study such systems with concepts and methods similar to those of standard statistical mechanics. The answer appears to be yes for ubiquitous systems, but the concept of entropy needs to be adequately generalized. Some classes of such systems can be satisfactorily approached with the entropy \(S_q = k\frac{1-\sum_{i = 1}^W p_i^q}{q-1}\) (with \(q \in \cal R\), and \(S_1 = S_{\rm BG}\)). This theory is sometimes referred in the literature as nonextensive statistical mechanics. We provide here a brief introduction to the formalism, its dynamical foundations, and some illustrative applications. In addition to these, we illustrate with a few examples the concept of stability (or experimental robustness) introduced by B. Lesche in 1982 and recently revisited by S. Abe.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

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 

Download references

Author information

Authors and Affiliations

  1. Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro-RJ, Brazil

    C. Tsallis & E. Brigatti

Authors
  1. C. Tsallis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Brigatti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Tsallis.

Additional information

Communicated by M. Sugiyama

Received: 27 May 2003, Accepted: 27 August 2003, Published online: 11 February 2004

PACS:

05.10.-a, 05.20.Gc, 05.20.-y, 05.45.-a, 05.70.Ln, 05.90. + m, 05.60.Cd

Correspondence to: C. Tsallis

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tsallis, C., Brigatti, E. Nonextensive statistical mechanics: A brief introduction. Continuum Mech. Thermodyn. 16, 223–235 (2004). https://doi.org/10.1007/s00161-004-0174-4

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-004-0174-4

Keywords:

Search

Navigation