Criticality in non-linear one-dimensional maps: RG universal map and non-extensive entropy

A. Robledo

doi:10.1016/j.physd.2004.01.016

paper

Criticality in non-linear one-dimensional maps: RG universal map and non-extensive entropy

arXiv:cond-mat/0202095 · doi:10.1016/j.physd.2004.01.016

Abstract

We consider the period-doubling and intermittency transitions in iterated nonlinear one-dimensional maps to corroborate unambiguously the validity of Tsallis' non-extensive statistics at these critical points. We study the map $x_{n + 1} = x_{n} + u ∣ x_{n} ∣^{z}$ , $z > 1$ , as it describes generically the neighborhood of all of these transitions. The exact renormalization group (RG) fixed-point map and perturbation static expressions match the corresponding expressions for the dynamics of iterates. The time evolution is universal in the RG sense and the non-extensive entropy $S_{Q}$ associated to the fixed-point map is maximum with respect to that of the other maps in its basin of attraction. The degree of non-extensivity - the index $Q$ in $S_{Q}$ - and the degree of nonlinearity $z$ are equivalent and the generalized Lyapunov exponent $λ_{q}$ , $q = 2 - Q^{- 1}$ , is the leading map expansion coefficient $u$ . The corresponding deterministic diffusion problem is similarly interpreted. We discuss our results.

To be published in Physica D, expanded version, updated references

Criticality in non-linear one-dimensional maps: RG universal map and non-extensive entropy

Abstract

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