Diffusive dynamics in systems with translational symmetry: A one-dimensional-map model

Schell, Mark; Kapral, Raymond

doi:10.1103/physreva.26.504

Cited by 121 publications

(94 citation statements)

References 22 publications

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“…These ballistic solutions are born through tangent bifurcations, further undergo a Feigenbaum-type scenario and die at crises points [6,8]. Localized solutions occurr at even periods only and start with tangent bifurcations followed by a symmetry breaking at slope-type bifurcation points [6,8].…”

Section: Periodic Windowsmentioning

confidence: 99%

“…Under parameter variation these different types of dynamics are highly intertwined resulting in complicated scenarios related to the appearance of periodic windows [6,9]. In order to study diffusion we will be interested in parameters that are greater than a > a 0 = 0.732644... for which the extrema of the map exceed the boundaries of each box for the first time indicating the onset of diffusive motion.…”

Section: The Climbing Sine Mapmentioning

confidence: 99%

“…We first approximate the invariant density in Eq. (10) to ρ(x) ≃ 1 irrespective of the fact that it is typically a very complicated function of x and a [6,38].…”

Section: Fine Scale Of the Diffusion Coeffcient: Anomalous Diffusion mentioning

confidence: 99%

“…In the limiting case of strong dissipation they can be reduced to nonhyperbolic onedimensional time-discrete maps sharing certain symmetries [53,54]. The socalled climbing sine map is a well-known example of this class of maps [5,6,9].…”

Section: Introductionmentioning

confidence: 99%

“…The analysis of these simple models required to suitably combine nonequilibrium statistical mechanics with dynamical systems theory leading to a more profound understanding of transport in nonequilibrium situations [1,2,3,4]. However, time-discrete maps provide not only a suitable starting point for studying normal diffusion but also for investigating the anomalous case [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. Moreover, there are certain classes of more realistic models which share specific properties of maps such as being low-dimensional and exhibiting certain periodicities.…”

Section: Introductionmentioning

confidence: 99%

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Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Korabel

Klages

2004

Physica D: Nonlinear Phenomena

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The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameterdependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green-Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.

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Section: Periodic Windowsmentioning

confidence: 99%

Section: The Climbing Sine Mapmentioning

confidence: 99%

“…We first approximate the invariant density in Eq. (10) to ρ(x) ≃ 1 irrespective of the fact that it is typically a very complicated function of x and a [6,38].…”

Section: Fine Scale Of the Diffusion Coeffcient: Anomalous Diffusion mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Korabel

Klages

2004

Physica D: Nonlinear Phenomena

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Deterministic (Anomalous) Transport

Artuso

Cristadoro

2008

Anomalous Transport

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Excitation Propagation in the Presence of a Driven and Kicked Vibration

Esser

1990

Physica Status Solidi (b)

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The propagation of excitons interacting with a driven and kicked vibration is considered using the generalized master equation (GME) approach. The amplitude of the vibration is described by a nonlinear evolution equation with a kicking term modelling a periodic perturbation. The connection between the GME kernel (memory function) and a map describing the dynamics of the vibration is established. This connection is used to discuss the behaviour of the GME kernel and resulting exciton motion following from the periodic and chaotic solutions of the map. For chaotic solutions an ensemble is introduced and the decay of the memory function calculated.Ausgehend von der generalisierten Master-Gleichung (GME) wird die Ausbreitung von Exzitonen, die mit einer gepumpten und angestoeenen Schwingung wechselwirken, betrachtet. Die Schwingungsamplitude wird durch eine nichtlineare Evolutionsgleichung beschrieben, die einen StoRterm enthalt, der eine periodische Storung modelliert. E s wird eine Verbindung des Kernes der GME mit einer diskreten Abbildung, die die Dynamik der Schwingung beschreibt, hergestellt. Diese Verbindung wird benutzt, um das aus den periodischen und chaotischen Losungen der Abbildung folgende Verhalten des GME-Kernes und der exzitonischen Bewegung zu diskutieren. Fur chaotische Losungen wird ein Ensemble eingefuhrt und das Abklingen des Kernes berechnet.

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Diffusive dynamics in systems with translational symmetry: A one-dimensional-map model

Cited by 121 publications

References 22 publications

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Deterministic (Anomalous) Transport

Excitation Propagation in the Presence of a Driven and Kicked Vibration

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