Vanishing largest Lyapunov exponent and Tsallis entropy

Abstract

We present a geometric argument that explains why some systems having vanishing largest Lyapunov exponent have underlying dynamics aspects of 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 about systems that have a simple asymptotic behaviour and are described by the Tsallis entropy.

15 pages, No figures. LaTex2e. Some overlap with arXiv:1104.4869 Additional references and clarifications in this version. To be published in QScience Connect

References in corpus (12)

• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Numerical indications of a q-generalised central limit theorem
• A note on q-Gaussians and non-Gaussians in statistical mechanics
• Influence of global correlations on central limit theorems and entropic extensivity
• Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos
• Distributivity and deformation of the reals from Tsallis entropy
• Tsallis entropy induced metrics and CAT(k) spaces
• Sensitivity, entropy, and escape rates at the onset of chaos
• Some Open Points in Nonextensive Statistical Mechanics
• Two-dimensional dissipative maps at chaos threshold: Sensitivity to initial conditions and relaxation dynamics
• Long memory constitutes a unified mesoscopic mechanism consistent with nonextensive statistical mechanics
• Weak Chaos from Tsallis Entropy

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