Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos

Ernesto P. Borges; Constantino Tsallis; Garín F.J. Añaños; Paulo Murilo C. de Oliveira

doi:10.1103/PhysRevLett.89.254103

Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos

Ernesto P. Borges, Constantino Tsallis, Garín F.J. Añaños, Paulo Murilo C. de Oliveira

Research output: Contribution to journal › Article › peer-review

102 Scopus citations

Abstract

We consider nonequilibrium probabilistic dynamics in logisticlike maps [Formula presented], [Formula presented] at their chaos threshold: We first introduce many initial conditions within one among [Formula presented] intervals partitioning the phase space and focus on the unique value [Formula presented] for which the entropic form [Formula presented] linearly increases with time. We then verify that [Formula presented] vanishes like [Formula presented] [[Formula presented]]. We finally exhibit a new finite-size scaling, [Formula presented]. This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.

Original language	English
Journal	Physical Review Letters
Volume	89
Issue number	25
DOIs	https://doi.org/10.1103/PhysRevLett.89.254103
State	Published - 2002

Bibliographical note

Funding Information:
One of us (C. T.) thanks H. J. Herrmann for fruitful remarks. This work is partially supported by PRONEX/MCT, CNPq, CAPES, and FAPERJ (Brazilian agencies).

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title = "Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos",

abstract = "We consider nonequilibrium probabilistic dynamics in logisticlike maps [Formula presented], [Formula presented] at their chaos threshold: We first introduce many initial conditions within one among [Formula presented] intervals partitioning the phase space and focus on the unique value [Formula presented] for which the entropic form [Formula presented] linearly increases with time. We then verify that [Formula presented] vanishes like [Formula presented] [[Formula presented]]. We finally exhibit a new finite-size scaling, [Formula presented]. This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.",

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N1 - Funding Information: One of us (C. T.) thanks H. J. Herrmann for fruitful remarks. This work is partially supported by PRONEX/MCT, CNPq, CAPES, and FAPERJ (Brazilian agencies).

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N2 - We consider nonequilibrium probabilistic dynamics in logisticlike maps [Formula presented], [Formula presented] at their chaos threshold: We first introduce many initial conditions within one among [Formula presented] intervals partitioning the phase space and focus on the unique value [Formula presented] for which the entropic form [Formula presented] linearly increases with time. We then verify that [Formula presented] vanishes like [Formula presented] [[Formula presented]]. We finally exhibit a new finite-size scaling, [Formula presented]. This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.

AB - We consider nonequilibrium probabilistic dynamics in logisticlike maps [Formula presented], [Formula presented] at their chaos threshold: We first introduce many initial conditions within one among [Formula presented] intervals partitioning the phase space and focus on the unique value [Formula presented] for which the entropic form [Formula presented] linearly increases with time. We then verify that [Formula presented] vanishes like [Formula presented] [[Formula presented]]. We finally exhibit a new finite-size scaling, [Formula presented]. This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.

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Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos

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