Vanishing largest Lyapunov exponent and Tsallis entropy

TL;DR

This paper explains why systems with vanishing largest Lyapunov exponents can be effectively described by Tsallis entropy, using geometric arguments involving hyperbolic metrics and entropy additivity properties.

Contribution

It provides a geometric explanation linking vanishing Lyapunov exponents to Tsallis entropy through hyperbolic metrics and generalized additivity.

Findings

Hyperbolic metrics relate to vanishing Lyapunov exponents.

Tsallis entropy's generalized additivity contrasts with BGS entropy.

The approach aligns with known systems described by 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.