Fractals and dynamical chaos in a two-dimensional Lorentz gas with sinks

Claus, Isabelle; Gaspard, Pierre

doi:10.1103/physreve.63.036227

Cited by 17 publications

(27 citation statements)

References 34 publications

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“…These absorbing boundary conditions select a set of phase-space trajectories, forming a chaotic and fractal repeller, which is related to an equation for K-S entropy. The escape-rate formalism has applications in diffusion [27], reaction-diffusion [28] and, recently, viscosity [29]. Another application is the classification of quantum dynamical systems, which is given by Ohya [30].…”

Section: The Importance Of Entropymentioning

confidence: 99%

The Entropy of Co-Compact Open Covers

Zheng

Wang

Wei

et al. 2013

Entropy

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Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability are not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: (1) it does not require the space to be compact and, thus, generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems; and (2) it is an invariant of topological conjugation, compared to Bowen's entropy, which is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system, (R, f ), defined by f (x) = 2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More generally, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result also generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces.

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Section: The Importance Of Entropymentioning

confidence: 99%

The Entropy of Co-Compact Open Covers

Zheng

Wang

Wei

et al. 2013

Entropy

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“…Such systems can be characterized by the drift speed and diffusion coefficient. If particles can be lost from the point of view of diffusion by absorption, chemical reaction or escape in directions transverse to the extension of the system, we refer to transient diffusion [19][20][21]. We set A = ∆ and we obtain a shift density σ ∆ in analogy to the procedure above: σ ∆ = T L∆T ρ in .…”

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confidence: 99%

Comparison of averages of flows and maps

Kaufmann

Lustfeld

2001

Phys. Rev. E

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It is shown that in transient chaos there is no direct relation between averages in a continuos time dynamical system (flow) and averages using the analogous discrete system defined by the corresponding Poincaré map. In contrast to permanent chaos, results obtained from the Poincaré map can even be qualitatively incorrect. The reason is that the return time between intersections on the Poincaré surface becomes relevant. However, after introducing a true-time Poincaré map, quantities known from the usual Poincaré map, such as conditionally invariant measure and natural measure, can be generalized to this case. Escape rates and averages, e.g. Liapunov exponents and drifts can be determined correctly using these novel measures. Significant differences become evident when we compare with results obtained from the usual Poincaré map.Extensive investigations of chaotic systems in recent years have demonstrated the great importance of transient chaos, due mainly to its connection with transport phenomena [1-3] and chaotic advection [4], possibly associated with chemical reactions [5]. In most chaotic systems it is sufficient to know the intersection points of the trajectories with a chosen surface P , the so-called Poincaré surface. In case of N -dimensional phase space P is N − 1-dimensional. Using a coordinate system on P , and finding the connection between the successive intersections x n and x n+1 the Poincaré map (P M ) can be constructed asThe behavior of the system can then be studied by iteration of this map. The advantages of the use of P M are (i) it is discrete, (ii) it has smaller dimension. Its disadvantage is the absence of the close connection between the number of intersections n and the time t, since the return time τ between two intersections depends generically on where a trajectory intersects. One can keep this information by completing the P M with the equationWe call this extended map the true-time Poincaré map (T P M ). Usually, one reduces to the P M by the following argument: The total time after n iterations is given by the sum of the corresponding return times τ (x). It is generally assumed that for large n and for typical trajectories the terms in the sum can be replaced by their average over the invariant density ρ P of the map. The sum then becomes a product [6] Based on this connection, averages of the map (using n for time) and the flow (using the real time t) would be simply related by a time scale. This is explicitly shown for general averages in case of permanent (non-transient) chaos [7]. We demonstrate in this paper that, in contrast to permanent chaos, the situation is quite different for transient chaos. Not only τ in Eq. (3) should be changed, but averages of the map and the flow (or of the T P M representing it) are not any more related by a time scale. The situation is somewhat reminiscent of the case when, instead of simple averages, the decay rates of correlations are considered. Even in permanent chaos, these show a discrepancy in non-ideal situations [8]. To proceed in a c...

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“…The escape rate characterizing the exponential decay may be obtained as follows [4,5,6]. Let us take N 0 initial conditions randomly distributed in the phase space of variables Γ = (x, y, ψ) according to some initial measure ν 0 .…”

Section: Escape and Fractal Repeller 221 Asymptotic Behaviormentioning

confidence: 99%

“…It can therefore be studied by the methods developed in Refs. [4,5,6], in order to relate its microscopic chaotic dynamics to its macroscopic exponential decay.…”

Section: Fractal Repeller and Nonequilibrium Probability Measurementioning

confidence: 99%

“…The fractal repeller associated with the deterministic chaotic diffusion in the presence of periodically distributed traps or sinks has been studied in a previous paper [6]. The system studied there was a two-dimensional periodic Lorentz gas: a point particle undergoes elastic collisions on hard disks fixed in the plane and forming a regular triangular lattice.…”

Section: Introductionmentioning

confidence: 99%

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Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

Claus

Gaspard

Beijeren

2004

Physica D: Nonlinear Phenomena

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Two-dimensional random Lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap. In systems of finite spatial extension, the asymptotic decay of the survival probability is exponential and characterized by an escape rate γ, which can be related to the average positive Lyapunov exponent and to the dimension of the fractal repeller of the system. For infinite systems, the survival probability obeys a stretched exponential law of the form P (c, t) ∼ exp(−Ct 1/2 ).The transition between the two regimes is studied and we show that, for a given trap density, the non-integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space. Nevertheless, the repeller remains fractal. We determine the special scaling properties of this fractal.

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Fractals and dynamical chaos in a two-dimensional Lorentz gas with sinks

Cited by 17 publications

References 34 publications

The Entropy of Co-Compact Open Covers

The Entropy of Co-Compact Open Covers

Comparison of averages of flows and maps

Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

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