Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points

Robledo, Á.

doi:10.1016/j.physa.2006.06.003

Cited by 33 publications

(58 citation statements)

References 45 publications

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“…A critical discussion of this approach is given in Ref. [39] (and see a reply in [40]). Note that in this case the invariant density is a discontinuous fractal Cantor set [36] which is different from the absolutely continuous but infinite invariant densities inherent in our approach.…”

Section: Introductionmentioning

confidence: 99%

Separation of trajectories and its relation to entropy for intermittent systems with a zero Lyapunov exponent

Korabel

Barkai

2010

Phys. Rev. E

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One dimensional intermittent maps with stretched exponential δxt ∼ δx0e λαt α separation of nearby trajectories are considered. When t → ∞ the standard Lyapunov exponent λ = t−1 i=0 ln |M ′ (xi)| /t is zero (M ′ is a Jacobian of the map). We investigate the distribution ofwhere α is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of λα is determined by the infinite invariant density. Using semi analytical arguments we calculate the infinite invariant density for the PomeauManneville map, and with it obtain excellent agreement between numerical simulation and theory. We show that α λα is equal to Krengel's entropy and to the complexity calculated by the LempelZiv compression algorithm. This generalized Pesin's identity shows that λα and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.

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Section: Introductionmentioning

confidence: 99%

Separation of trajectories and its relation to entropy for intermittent systems with a zero Lyapunov exponent

Korabel

Barkai

2010

Phys. Rev. E

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“…A critical discussion of this approach is given in Ref. [14] (and see a reply in [15]). According to [14] a meaningful generalized Pesin's identity must satisfy certain requirements.…”

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confidence: 99%

Pesin-Type Identity for Intermittent Dynamics with a Zero Lyaponov Exponent

2009

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Pesin's identity provides a profound connection between entropy hKS (statistical mechanics) and the Lyapunov exponent λ (chaos theory). It is well known that many systems exhibit sub-exponential separation of nearby trajectories and then λ = 0. In many cases such systems are non-ergodic and do not obey usual statistical mechanics. Here we investigate the non-ergodic phase of the PomeauManneville map where separation of nearby trajectories follows δxt = δx0e λαt α with 0 < α < 1. The limit distribution of λα is the inverse Lévy function. The average λα is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.PACS numbers: 05.90.+m, 05.45.Ac, 74.40.+k Chaotic systems are characterized by exponential separation of nearby trajectories, which is quantified by a positive Lyapunov exponent λ [1]. Such a behavior leads to the need for statistical approaches since chaos implies our inability to predict the long time limit of a system in a deterministic fashion. A profound relation between chaos and statistical mechanics is given by Pesin's identity [1]. It states that the Kolmogorov-Sinai entropy h KS is equal to the Lyapunov exponent λ for a closed ergodic 1d systems (to the sum of positive Lyapunov exponents for d > 1). At the same time, it is well known that many systems such as Hamiltonian models with mixed phase space [2], systems with long range forces [3], certain billiards [4] and one-dimensional hard-particle gas [5] have a Lyapunov exponent equal zero. While for complex systems it may be extremely difficult to determine whether the Lyapunov exponent is zero or small, due to numerical inaccuracies, it turns out that most fundamental text book examples of chaos theory may have a zero Lyapunov exponent. Prominent examples for such weakly chaotic systems are the logistic map at the edge of chaos (Feigenbaum's point) [6] and the Pomeau-Manneville map which is used to model intermittency (originally in turbulence) [7]. If the Lyapunov exponent is zero, i.e. separation of trajectories is sub-exponential, we have a strong indication that the usual Boltzmann-Gibbs statistical mechanics is not valid. Indeed it was found that certain systems with zero Lyapunov exponents break ergodicity [8,9]. Classical entropy theory is also not applicable in this case [6,7], particularly the entropy and average algorithmic complexity grow non linearly in time [7], while for a system with a positive Lyapunov exponent they increase linearly in time. Still the situation is not hopeless from the point of view of statistical mechanics and one may consider distributions of time average observables [8,9,10,11,12].Connection between possible generalizations of usual statistical mechanics and weakly chaotic systems have attracted much attention recently. In particular, a generalized Pesin's identity for the logistic map at the edge of chaos was investigated using Tsallis statistics [6,13]. A critical discussion of this appr...

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“…However, the change in the dynamics so as to make the stationary distributions of the Van der Pol model of q-exponential form, results in a spectrum of some privileged q values. This seems to be a general feature one encounters whenever one studies physical systems depending on a control parameter [27,28]. It is also worth remarking how the probability distribution of the first constraint in nonadditive thermostatistics i.e., q-exponential with order (2−q), emerges in the numerical studies concerning systems depending upon control parameters [27,28].…”

Section: Discussionmentioning

confidence: 82%

Self-Organization in Nonadditive Systems With External Noise

Bagci

Tirnakli

2009

Int. J. Bifurcation Chaos

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A nonadditive generalization of Klimontovich's S-theorem [G. B. Bagcı, Int.J. Mod. Phys. B 22, 3381 (2008)] has recently been obtained by employing Tsallis entropy. This general version allows one to study physical systems whose stationary distributions are of the inverse power law in contrast to the original S-theorem, which only allows exponential stationary distributions. The nonadditive S-theorem has been applied to the modified Van der Pol oscillator with inverse power law stationary distribution. By using nonadditive S-theorem, it is shown that the entropy decreases as the system is driven out of equilibrium, indicating self-organization in the system. The allowed values of the nonadditivity index q are found to be confined to the regime (0.5, 1.0].

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Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points

Cited by 33 publications

References 45 publications

Separation of trajectories and its relation to entropy for intermittent systems with a zero Lyapunov exponent

Separation of trajectories and its relation to entropy for intermittent systems with a zero Lyapunov exponent

Pesin-Type Identity for Intermittent Dynamics with a Zero Lyaponov Exponent

Self-Organization in Nonadditive Systems With External Noise

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