1887
Volume 2013, Issue 1
  •  E-ISSN:  Will be obtained soon 2223-506X

Abstract

We present a geometric argument that explains why some systems having vanishing largest Lyapunov exponent have underlying dynamic aspects which can be effectively described by the Tsallis entropy. We rely on a comparison of the generalised additivity of the Tsallis entropy versus the ordinary additivity of the BGS entropy. We translate this comparison, in metric terms, by using an effective hyperbolic metric on the configuration/phase space for the Tsallis entropy versus the Euclidean one in the case of the BGS entropy. Solving the Jacobi equation for such hyperbolic metrics effectively sets the largest Lyapunov exponent computed, with respect to the corresponding Euclidean metric, to zero. This conclusion is in agreement with all currently known results on systems that have a simple asymptotic behaviour, and are described by the Tsallis entropy.

Loading

Article metrics loading...

/content/journals/10.5339/connect.2013.26
2013-12-01
2020-06-06
Loading full text...

Full text loading...

/deliver/fulltext/connect/2013/1/connect.2013.26.html?itemId=/content/journals/10.5339/connect.2013.26&mimeType=html&fmt=ahah

References

  1. Tsallis C. Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys. 1988; 52:1–2:479487.
    [Google Scholar]
  2. Mariz AM, Tsallis C. Unified long-memory mesoscopic mechanism consistent with nonextensive statistical mechanics. Phys Lett A. 2012; 376:45:30883091.
    [Google Scholar]
  3. Tsallis C. Introduction to Nonextensive Statistical Mechanics. : Springer-Verlag 2009.
    [Google Scholar]
  4. Boltzmann L. Akademie der Wissenschaften in Wien. Math Natur Kl. 1877; 75::67.
    [Google Scholar]
  5. Gallavotti G. Statistical Mechanics: A Short Treatise. : Springer-Verlag 1999.
    [Google Scholar]
  6. Cohen EGD. Boltzmann and einstein: statistics and dynamics – an unsolved problem. Pramana. 2005; 64:5:635643.
    [Google Scholar]
  7. Grassberger P, Scheunert M. Some more universal scaling laws for critical mappings. J Stat Phys. 1981; 26:4:697717.
    [Google Scholar]
  8. Anania G, Politi A. Dynamical behaviour at the onset of chaos. Europhys Lett. 1988; 7:2:119124.
    [Google Scholar]
  9. Hata H, Horita T, Mori H. Dynamic description of the critical 2 attractor and 2m-band chaos. Prog Theor Phys. 1989; 82:5:897910.
    [Google Scholar]
  10. Fuentes MA, Sato Y, Tsallis C. Sensitivity to initial conditions, entropy production, and escape rate at the onset of chaos. Phys Lett A. 2011; 375:33:29882991.
    [Google Scholar]
  11. Tsallis C. Some open points in nonextensive statistical mechanics. Int J Bifurcat Chaos. 2012; 22:09:1230030.
    [Google Scholar]
  12. Kalogeropoulos N. Weak chaos from Tsallis entropy. QScience Connect. 2012; 2012:12.
    [Google Scholar]
  13. Radhakrishna Rao C. Information and accuracy attainable in the estimation of statistical parameters. Bull Calcutta Math Soc. 1945; 37::8191.
    [Google Scholar]
  14. Csiszar I. I-divergence geometry of probability distributions and minimization problems. Ann Probab. 1975; 3:1:146158.
    [Google Scholar]
  15. Efron B. Defining the curvature of a statistical problem (with applications to second order efficiency). Ann Stat. 1975; 3:6:11891242.
    [Google Scholar]
  16. Amari S-I. Differential geometry of curved exponential families-curvatures and information loss. Ann Stat. 1982; 10:2:357385.
    [Google Scholar]
  17. Amari SI. Differential-Geometrical Methods in Statistics. : Springer-Verlag 1985.
    [Google Scholar]
  18. Barndorff-Nielsen OE. Differential geometry and statistics: some mathematical aspects. Indian J Math. 1987; 29:3:335350.
    [Google Scholar]
  19. Amari SI, Nagaoka H. Translations of Mathematical Monographs: Methods of Information Geometry. American Mathematical Society, Oxford University Press 2000:p.191.
    [Google Scholar]
  20. Rachev ST. Probability Metrics and the Stability of Stochastic Models. : Wiley 1991.
    [Google Scholar]
  21. Gromov M. Metric Structures for Riemannian and Non-Riemannian Spaces. Modern Birkhauser Classics: Birkhauser 1999.
    [Google Scholar]
  22. Tsallis C, Plastino AR, Zheng W-M. Power-law sensitivity to initial conditions—New entropic representation. Chaos, Solitons Fract. 1997; 8:6:885891.
    [Google Scholar]
  23. Costa U, Lyra M, Plastino A, Tsallis C. Power-law sensitivity to initial conditions within a logisticlike family of maps: fractality and nonextensivity. Phys Rev E. 1997; 56:1:245250.
    [Google Scholar]
  24. Lyra M, Tsallis C. Nonextensivity and multifractality in low-dimensional dissipative systems. Phys Rev Lett. 1998; 80:1:5356.
    [Google Scholar]
  25. Anteneodo C, Tsallis C. Breakdown of exponential sensitivity to initial conditions: role of the range of interactions. Phys Rev Lett. 1998; 80:24:53135316.
    [Google Scholar]
  26. Trnakl U, Tsallis C, Lyra ML. Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity. Eur Phys J B. 1999; 11:2:309315.
    [Google Scholar]
  27. Latora V, Baranger M, Rapisarda A, Tsallis C. The rate of entropy increase at the edge of chaos. Phys Lett A. 2000; 273:1-2:97103.
    [Google Scholar]
  28. Borges E, Tsallis C, Aaos G, de Oliveira P. Nonequilibrium probabilistic dynamics of the logistic map at the edge of chaos. Phys Rev Lett. 2002; 89:25.
    [Google Scholar]
  29. Baldovin F, Robledo A. Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions. Europhys Lett. 2002; 60:4:518524.
    [Google Scholar]
  30. Baldovin F, Robledo A. Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics. Phys Rev E. 2002; 66:4.
    [Google Scholar]
  31. Baldovin F, Tsallis C, Schulze B. Nonstandard entropy production in the standard map. Phys A Stat Mech Appl. 2003; 320::184192.
    [Google Scholar]
  32. Aaos G, Tsallis C. Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps. Phys Rev Lett. 2004; 93:2.
    [Google Scholar]
  33. Borges EP, Tirnakli U. Two-dimensional dissipative maps at chaos threshold: sensitivity to initial conditions and relaxation dynamics. Phys A Stat Mech Appl. 2004; 340:1–3:227233.
    [Google Scholar]
  34. Aaos GFJ, Baldovin F, Tsallis C. Anomalous sensitivity to initial conditions and entropy production in standard maps: nonextensive approach. Eur Phys J B. 2005; 46:3:409417.
    [Google Scholar]
  35. Celikoglu A, Tirnakli U. Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos. Phys A Stat Mech Appl. 2006; 372:2:238242.
    [Google Scholar]
  36. Tirnakli U, Tsallis C. Chaos edges of z-logistic maps: connection between the relaxation and sensitivity entropic indices. Phys Rev E. 2006; 73:3.
    [Google Scholar]
  37. Moyano LG, Tsallis C, Gell-Mann M. Numerical indications of a q-generalised central limit theorem. Europhys Lett. 2006; 73:6:813819.
    [Google Scholar]
  38. Marsh JA, Fuentes MA, Moyano LG, Tsallis C. Influence of global correlations on central limit theorems and entropic extensivity. Phys A Stat Mech Appl. 2006; 372:2:183202.
    [Google Scholar]
  39. Tsallis C. Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem. Phys A Stat Mech Appl. 2006; 365:1:716.
    [Google Scholar]
  40. Hilhorst HJ, Schehr G. A note on q-Gaussians and non-Gaussians in statistical mechanics. J Stat Mech Theor Exp. 2007; 2007:06:P06003.
    [Google Scholar]
  41. Rodrguez A, Schwmmle V, Tsallis C. Strictly and asymptotically scale invariant probabilistic models of N correlated binary random variables having q-Gaussians as N limiting distributions. J Stat Mech Theor Exp. 2008; 2008:09:P09006.
    [Google Scholar]
  42. Kalogeropoulos N. Tsallis entropy induced metrics and spaces. Phys A Stat Mech Appl. 2012; 391:12:34353445.
    [Google Scholar]
  43. Cheeger J, Ebin DG. Comparison Theorems in Riemannian Geometry. : AMS Chelsea 1975.
    [Google Scholar]
  44. Sakai T. Translations of Mathematical Monographs: Riemannian Geometry. : American Mathematical Society, Oxford University Press 1996:p.149.
    [Google Scholar]
  45. Bridson MR, Haefliger A. Metric Spaces of Non-Positive Curvature. Fundamental Principles of Mathematical Sciences, : Springer-Verlag 1999:p.319.
    [Google Scholar]
  46. Nivanen L, Le Mhaut A, Wang QA. Generalized algebra within a nonextensive statistics. Rep Math Phys. 2003; 52:3:437444.
    [Google Scholar]
  47. Borges EP. A possible deformed algebra and calculus inspired in nonextensive thermostatistics. Phys A Stat Mech Appl. 2004; 340:1–3:95101.
    [Google Scholar]
  48. Petit Loba o TC, Cardoso PGS, Pinho STR, Borges EP. Some properties of deformed q-numbers. Braz J Phys. 2009; 39:2A:402.
    [Google Scholar]
  49. Kalogeropoulos N. Distributivity and deformation of the reals from Tsallis entropy. Phys A Stat Mech Appl. 2012; 391:4:11201127.
    [Google Scholar]
  50. In: Ghys Ede la Harpe P, eds. Sur les Groupes Hyperboliques d'après Mikhael Gromov. Progress in Mathematics, : Birkhäuser 1990.
    [Google Scholar]
  51. Katok A, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, : Cambridge University Press 1995:p.54.
    [Google Scholar]
  52. Barreira L, Pesin Y. Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Encyclopedia of Mathematics and its Applications, : Cambridge University Press 2007:p.115.
    [Google Scholar]
  53. Gersten SM. Quadratic Divergence of Geodesics in CAT(0) Spaces. Geom Funct Anal. 1994; 4:1:3751.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.5339/connect.2013.26
Loading
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error