Equilibration of energy in slow–fast systems Shah, Kushal; Turaev, Dmitry; Gelfreich, Vassili ...
Proceedings of the National Academy of Sciences - PNAS,
12/2017, Letnik:
114, Številka:
49
Journal Article
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Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. However, in systems with several characteristic timescales, the ergodicity of the fast ...subsystem impedes the equilibration of the whole system because of the presence of an adiabatic invariant. In this paper, we show that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium. To show this principle, we investigate the dynamics of springy billiards, which are mechanical systems composed of a small particle bouncing elastically in a bounded domain, where one of the boundary walls has finite mass and is attached to a linear spring. Numerical simulations show that the springy billiard systems approach equilibrium at an exponential rate. However, in the limit of vanishing particle-to-wall mass ratio, the equilibration rates remain strictly positive only when the fast particle dynamics reveal two or more ergodic components for a range of wall positions. For this case, we show that the slow dynamics of the moving wall can be modeled by a random process. Numerical simulations of the corresponding springy billiards and their random models show equilibration with similar positive rates.
It is well known that typical Hamiltonian systems have divided phase space consisting of regions with regular dynamics on KAM tori and region(s) with chaotic dynamics called chaotic sea(s). This ...complex structure makes rigorous analysis of such systems virtually impossible and significantly complicates numerical exploration of their dynamical properties. Hamiltonian systems with sharply divided phase space between regions of regular and chaotic dynamics are much easier to analyze, but there are only few cases or families of such systems known to date. In this article, we outline a new approach for a systematic construction, starting from a generic KAM Hamiltonian system, of a system with a sharply divided phase space with an arbitrary number of regular islands which are in one-to-one correspondence with islands of the initial KAM system. In this procedure a typical Hamiltonian system, for example a KAM billiard, is replaced by a sequence of Hamiltonian systems having an increasing (but finite) number of islands of regular motion. The islands in the substituting systems are sub-islands of the KAM islands in the initial system. We apply this idea to two-dimensional lemon-shaped billiards, where the substituting systems are obtained by replacing parts of the curved boundaries by chords, so that in the limit of infinite number of islands the boundary of the substituting system becomes arbitrary close to the original billiard's boundary.
Hamiltonian systems typically exhibit a mixture of chaos and regularity, complicating any scheme to partition phase space and extract a symbolic description of the dynamics. In particular, the ...dynamics in the vicinity of stable islands can exhibit complicated topology that is qualitatively distinct from that away from the islands. We develop an approach to partition the chaotic phase space of a general dynamical system represented by a two-dimensional map (homeomorphism). This approach can accommodate mixed phase space structure with an arbitrarily high degree of accuracy. The partitioning scheme is built around networks of nested heteroclinic tangles—fundamental geometric objects that organize phase–space transport. These tangles can be used to progressively approximate the dynamics in the vicinity of stable island chains. The net result is a symbolic approximation to the dynamics in the chaotic sea, and an associated phase–space partition, which includes the influence of stable islands and which is approximately Markov.
► We present a method to partition the phase space of a chaotic 2D map. ► This method can analyze a map with stable islands embedded within a chaotic sea. ► The partition is constructed from networks of nested homo/heteroclinic tangles. ► The resulting partition is associated with a Markov shift. ► The partition may be refined with arbitrary topological precision.
Homoclinic and heteroclinic tangles are fundamental phase space structures that help in organizing the transport defined by a chaotic map on a two-dimensional phase space. Previous work introduced ...the technique of “homotopic lobe dynamics” to study the topology of a homoclinic tangle. The present work generalizes homotopic lobe dynamics to describe the topology of a network of nested homoclinic and/or heteroclinic tangles. Using this theory, we demonstrate that the topological analysis of a network of nested tangles can lead to an efficient symbolic description of a chaotic phase space, even in the vicinity of islands of stability. By increasing the amount of topological information used (e.g. by introducing additional tangles at finer scales), this technique suggests an efficient route for defining accurate symbolic dynamics for phase spaces that exhibit a mixture of chaos and regularity.
Progress in the modelling of ray-based dynamics in large step-index fibres (SIF) is reported. In the model developed the propagation of a light distribution through the fibre is achieved by ...discretising the mixed phase space and employing an iterative finite operator. The model allows for the deformation of the core and/or cladding cross-sectional boundaries of the SIF. In the presence of such deformations the light distribution in the mixed phase space reveals the relationship between the input light distribution and the net ray flux entering the core.
We report experimental study on the evolution of the emission mechanism by using a phase-space imaging technique based on the local angular momentum conservation in emission process in a chaotic ...optical microcavity as the cavity deformation corresponding to nonintegrability increases. In the regime of weak deformation, we could image the output beam being emitted tangentially from a location at a certain distance away from the cavity boundary, indicating an evanescent tunneling emission. As the deformation increases further, the location of emission approaches the cavity boundary and the directionality starts to deviate from the tangential direction due to a refractive emission via chaos-assisted tunneling. Here is reported the construction of the internal phase space associated with the modes involved in the tunneling emission of light in the classical mixed phase space where regular and chaotic dynamics coexist.
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