Chaotic Transport Research Articles

Integer moments of complex Wishart matrices and Hurwitz numbers

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large- N expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O’Connell, and N. J. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys. 369 (2019), no. 3, 1091–1145] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the generating functions of monotone and strictly monotone Hurwitz numbers. The main result resolves the integrality conjecture formulated in [F. D. Cunden, F. Mezzadri, N. J. Simm, and P. Vivo, Correlators for the Wigner–Smith time-delay matrix of chaotic cavities, J. Phys. A 49 (2016), no. 18, 18LT01, 20 pp] on the time-delay cumulants in quantum chaotic transport. The precise combinatorial description of the cumulants given here may cast new light on the concordance between random matrix and semiclassical theories.

Semiclassical treatment of quantum chaotic transport with a tunnel barrier

We consider the problem of a semiclassical description of quantum chaotic transport, when a tunnel barrier is present in one of the leads. Using a semiclassical approach formulated in terms of a matrix model, we obtain transport moments as power series in the reflection probability of the barrier, whose coefficients are rational functions of the number of open channels M. Our results are therefore valid in the quantum regime and not only when M ≫ 1. The expressions we arrive at are not identical with the corresponding predictions from random matrix theory, but are in fact much simpler. Both theories agree as far as we can test.

Experimental Observations of Perturbed Rayleigh-Benard Convection and Analysis of Chaotic Fluid Transport

Experimental Observations of Perturbed Rayleigh-Benard Convection and Analysis of Chaotic Fluid Transport

Dynamical Chaos and Lateral Transport of a Passive Scalar in the Annular Reverse Jet Flow

The transition to dynamical chaos and the related lateral (cross-flow) transport of a passive scalar in the reverse annular jet flow generating two chains of wave-vortex structures are studied. The quasi-geostrophic equations for the barotropic (quasi-two-dimensional) flow written in polar coordinates with allowance for the beta-effect and external friction are solved numerically using a pseudospectral method. The critical parameters of the equilibrium flow with a complex “two-hump” azimuth velocity profile facilitating a faster transition to the complex dynamics are determined. Two regular multiharmonic regimes of wave generation are revealed with increasing flow supercriticality before the onset of Eulerian chaos. The occurrence of the complex flow dynamics is confirmed by a direct calculation of the largest Lyapunov exponent. The evolution of streamline images is analyzed by making video, thereby chains with single and composite structures are distinguished. The wavenumber-frequency spectra confirming the possibility of chaotic transport of the passive scalar are drawn for the basic regimes of wave generation. The power law exponents for the azimuth particle displacement and their variance, which proved the occurrence of the anomalous azimuth transport of the passive scalar, are determined. Lagrangian chaos is studied by computing the finite-time Lyapunov exponent and its distribution function. The internal chain (with respect to the annulus center) is found to be totally subject to Lagrangian chaos, while only the external chain boundary is chaotic. It is revealed that the cross-flow transport occurs only in the regime of Eulerian dynamical chaos, since there exists a barrier to it in the multiharmonic regimes. The images of fluid particles confirming the presence of lateral transport are obtained and their quantitative characteristics are determined.

The Tachrift channel-levée turbidite complexes (Tortonian) of the Taza-Guercif basin (South Rifian Corridor, NE Morocco)

ABSTRACT This contribution reports on the field mapping of 9 exceptionally well-exposed channel-levée complexes from Taza–Guercif Basin (NE Morocco), belonging to the Late Miocene Tachrift turbidite system. Separated from each another by hemipelagic marlstones, the mapped channel-levée complexes exhibit thicknesses in the range of 5–25 m. Four main sedimentary facies associations were mapped, including channel-fill sandstones, levée thin-bedded heterolithics, chaotic mass transport deposits, and hemipelagic marlstones. In addition, two end-member styles of channel-fills spatial stacking were recognized, reflecting different modes of channel belt development and/or location along the slope profile, namely: (a) a lateral-migration pattern, resulting from lateral migration of high-sinuosity levéed channel belts, as opposed to (b) a vertically stacked pattern, interpreted to reflect the vertical aggradation of levéed channel belts with a relatively low sinuosity. The geological map accompanying this contribution provides the basis for more in-depth sedimentological investigations on the channels of the Tachrift turbidite system.

Enhanced flow boiling in microchannels by incorporating multiple micro-nozzles and micro-pinfin fences

Phase-change heat transfer is a promising approach for high-power electronics cooling. However, the chaotic two-phase transport and dominant laminar flow in microchannels inhibits flow boiling performance. Thus, the ability to coordinate the two-phase transport in a highly favorable fashion would be highly desired. In this study, a new microchannel configuration by incorporating capillary micro-pinfin fences and multiple micro-nozzles has been proposed to sustain thin liquid film evaporation, promote mixing and global liquid supply simultaneously. A new boundary layer covered with thin liquid film is activated by the capillary micro-pinfin fences along the sidewalls of the channel. Additionally, the sustainable thin liquid film is maintained using capillary effect, which can promote the capillary-driven flow inside the gap between micro-pinfin fences and the sidewalls. As a result, significantly enhanced flow boiling has been achieved on both DI-water and HFE-7100. A critical heat flux (CHF) up to 944 W/cm2 has been demonstrated using DI-water at a mass velocity of 600 kg/m2 s, accounting for a 43% enhancement compared to the configuration of multiple micro-nozzles without capillary micro-pinfin fences. Compared to the base configuration of plain wall microchannels, the enhancement of CHF is over threefold at a mass velocity of 389 kg/m2s. Equally important, this new microchannel configuration works well on dielectric fluids, which is challenging to promote CHF due to their unfavorable thermophysical properties. A CHF of 287 W/cm2 with a noticeable enhancement of 56% has been achieved at a mass velocity of 2772 kg/m2 s on HFE-7100 at room temperature. Moreover, the enhancements of heat transfer coefficient (HTC) on HFE-7100 are more noticeable compared to that of DI-water. For example, 120% higher HTC at a moderate mass velocity of 693 kg/m2 s has been realized. All the enhancements are achieved without the escalating pressure drop.

Meteorite transport—Revisited II

AbstractIn Wisdom (2017), I presented new simulations of meteorite transport from the chaotic zones associated with major resonances in the asteroid belt: the ν6 secular resonance, the 3:1 mean motion resonance with Jupiter, and the 5:2 mean motion resonance with Jupiter. I found that the observed afternoon excess (the fact that approximately twice as many meteorites fall in the afternoon as in the morning) of the ordinary chondrites is consistent with chaotic transport from the 3:1 resonance, contradicting prior reports. Here I report an additional study of the transport of meteorites from ν6 secular resonance and the 3:1 mean motion resonance. I use an improved integration algorithm, and study the evolution of more particles. I confirm that the afternoon excess of the ordinary chondrites is consistent with transport from the 3:1 resonance.

Cross-field chaotic transport of electrons by E × B electron drift instability in Hall thruster

A model calculation is presented to characterize the anomalous cross field transport of electrons in a Hall thruster geometry. The anomalous nature of the transport is attributed to the chaotic dynamics of the electrons arising from their interaction with fluctuating unstable electrostatic fields of the electron cyclotron drift instability that is endemic in these devices. The electrons energy gain from those background waves leads to a significant increase in their temperature along the perpendicular direction T⊥/T∥∼4 and an enhanced cross field electron transport along the thruster axial direction. It is shown that the wave-particle interaction induces a mean velocity of the electrons along the axial direction, which is of the same order of magnitude as seen in experimental observations.

Global organization of three-dimensional, volume-preserving flows: Constraints, degenerate points, and Lagrangian structure.

Global organization of three-dimensional (3D) Lagrangian chaotic transport is difficult to infer without extensive computation. For 3D time-periodic flows with one invariant, we show how constraints on deformation that arise from volume-preservation and periodic lines result in resonant degenerate points that periodically have zero net deformation. These points organize all Lagrangian transport in such flows through coordination of lower-order and higher-order periodic lines and prefigure unique transport structures that arise after perturbation and breaking of the invariant. Degenerate points of periodic lines and the extended 3D structures associated with them are easily identified through the trace of the deformation tensor calculated along periodic lines. These results reveal the importance of degenerate points in understanding transport in one-invariant fluid flows.

Experimental Analysis of Chaotic Transport and Lagrangian Coherent Structures in Rayleigh-Benard Convection

Experimental Analysis of Chaotic Transport and Lagrangian Coherent Structures in Rayleigh-Benard Convection

Optical radiation induced chaotic dynamics of electrons in a uniform magnetic field

We investigate the effects of linearly polarized optical radiation on the cyclotron motion of an electron wave packet, considering the full quantum dynamics of the system. Analysis of the Landau-level (LL) spectrum reveals that only intra band cyclotron oscillation frequencies contribute to the effective oscillation frequency of the motion, whereas scattering between electron and hole Landau levels are forbidden. We find that the wave packet dynamics is significantly affected by varying the polarization direction of the electromagnetic radiation. The optical radiation is also affected by its interaction with electrons. Interestingly, we find that chaotic effects are induced by radiation in the dynamics of electron wave packet in an applied uniform magnetic field. Chaotic signatures in the dynamics are diagnosed by computing the relevant out-of-time-order correlation function and analyzed by using Poincaré maps. We attribute the appearance of such chaotic transport of electron wave packet to the nonlinear interaction between the optical radiation and internal cooperative oscillating mode produced by the interplay of relativistic (zitterbewegung) and cyclotron oscillations.

Regular-Orbit-Engineered Chaotic Photon Transport in Mixed Phase Space.

The dynamical evolution of light in asymmetric microcavities is of primary interest for broadband optical coupling and enhanced light-matter interaction. Here, we propose and demonstrate that the chaos-assisted photon transport can be engineered by regular periodic orbits in the momentum-position phase space of an asymmetric microcavity. Remarkably, light at different initial states experiences different evolution pathways, following either regular-chaotic channels or pure chaotic channels. Experimentally, we develop a nanofiber technique to accurately control the excitation position of light in the phase space. We find that the coupling to high-Q whispering gallery modes depends strongly on excitation in islands or chaotic sea, showing a good agreement with the theoretical prediction. The engineered chaotic photon transport has potential in light manipulation, broadband photonic devices, and phase-space reconstruction.

Chaotic transport of navigation satellites.

Navigation satellites are known from numerical studies to reside in a dynamically sensitive environment, which may be of profound importance for their long-term sustainability. We derive the fundamental Hamiltonian of Global Navigation Satellite System dynamics and show analytically that near-circular trajectories lie in the neighborhood of a Normally Hyperbolic Invariant Manifold (NHIM), which is the primary source of hyperbolicity. Quasicircular orbits escape through chaotic transport, regulated by NHIM's stable and unstable manifolds, following a power-law escape time distribution P(t)∼t-α, with α∼0.8-1.5. Our study is highly relevant for the design of satellite disposal trajectories, using manifold dynamics.

Pattern formation in a fully three-dimensional segregating granular flow.

Segregation patterns of size-bidisperse particle mixtures in a fully three-dimensional flow produced by alternately rotating a spherical tumbler about two perpendicular axes are studied over a range of particle sizes and volume ratios using both experiments and a continuum model. Pattern formation results from the interaction of size segregation with chaotic regions and nonmixing islands of the flow. Specifically, large particles in the flowing surface layer are preferentially deposited in nonmixing islands despite the effects of collisional diffusion and chaotic transport. The protocol-dependent structure of the unstable manifolds of the flow surrounding the nonmixing islands provides further insight into why certain segregation patterns are more robust than others.

Chaotic transport of electron wave packet in Weyl semimetal slab.

We theoretically study the quantum transport of an electron wave packet on the Fermi arcs and in the bulk of a Weyl semimetal slab. The numerical analysis of the dynamical equations obtained from the Heisenberg equation of motion reveals that the electron motion in the Weyl semimetal exhibits interesting unusual effects. In particular, signatures of chaotic behavior in the transport of the electron wave packet are observed that are diagnosed by the relevant out-of-time-order correlation function and are analyzed using Poincaré maps. We attribute the appearance of such chaotic transport of the electron wave packet to the interplay of Zitterbewegung and cyclotron oscillations in the Weyl semimetal slab. The chaotic nature of the electron transport is exhibited both along the Fermi arcs and in the bulk of the slab, depending strongly upon the spin orientation of the electron. In the presence of a magnetic field, both interband and intraband (cyclotron) frequencies contribute to the resulting oscillation frequency of the electron motion.

Feasibility of a Resonance-based Planet Nine Search

Abstract It has been proposed that mean-motion resonances (MMRs) between Planet Nine and distant objects of the scattered disk might inform the semimajor axis and instantaneous position of Planet Nine. Within the context of this hypothesis, the specific distribution of occupied MMRs largely determines the available constraints. Here we characterize the behavior of scattered Kuiper Belt objects arising in the presence of an eccentric Planet Nine (e 9 ∈ 0.1, 0.7), focusing on relative sizes of populations occupying particular commensurabilities. Highlighting the challenge of predicting the exact MMR of a given object, we find that the majority of resonant test particles have period ratios with Planet Nine other than those of the form P 9/P = N/1, N/2 . Taking into account the updated prior distribution of MMRs outlined in this work, we find that the close spacing of high-order resonances, as well as chaotic transport, preclude resonance-based Planet Nine constraints from current observational data.

Controlling chaotic spin-motion entanglement of ultracold atoms via spin-orbit coupling.

We study the spatially chaoticity-dependent spin-motion entanglement of a spin-orbit (SO) coupled Bose-Einstein condensate with a source of ultracold atoms held in an optical superlattice. In the case of phase synchronization, we analytically demonstrate that (a) the SO coupling (SOC) leads to the generation of spin-motion entanglement; (b) the area of the high-chaoticity parameter region inversely relates to the SOC strength which renormalizes the chemical potential; and (c) the high-chaoticity is associated with the lower chemical potential and the larger ratio of the short-lattice depth to the longer-lattice depth. Then, we numerically generate the Poincaré sections to pinpoint that the chaos probability is enhanced with the decrease in the SOC strength and/or the spin-dependent current components. The existence of chaos is confirmed by computing the corresponding largest Lyapunov exponents. For an appropriate lattice depth ratio, the complete stop of one of (or both) the current components is related to the full chaoticity. The results mean that the weak SOC and/or the small current components can enhance the chaoticity. Based on the insensitivity of chaos probability to initial conditions, we propose a feasible scheme to manipulate the ensemble of chaotic spin-motion entangled states, which may be useful in coherent atom optics with chaotic atom transport.

Using periodic orbits to compute chaotic transport rates between resonance zones.

Transport properties of chaotic systems are computable from data extracted from periodic orbits. Given a sufficient number of periodic orbits, the escape rate can be computed using the spectral determinant, a function that incorporates the eigenvalues and periods of periodic orbits. The escape rate computed from periodic orbits converges to the true value as more and more periodic orbits are included. Escape from a given region of phase space can be computed by considering only periodic orbits that lie within the region. An accurate symbolic dynamics along with a corresponding partitioning of phase space is useful for systematically obtaining all periodic orbits up to a given period, to ensure that no important periodic orbits are missing in the computation. Homotopic lobe dynamics (HLD) is an automated technique for computing accurate partitions and symbolic dynamics for maps using the topological forcing of intersections of stable and unstable manifolds of a few periodic anchor orbits. In this study, we apply the HLD technique to compute symbolic dynamics and periodic orbits, which are then used to find escape rates from different regions of phase space for the Hénon map. We focus on computing escape rates in parameter ranges spanning hyperbolic plateaus, which are parameter intervals where the dynamics is hyperbolic and the symbolic dynamics does not change. After the periodic orbits are computed for a single parameter value within a hyperbolic plateau, periodic orbit continuation is used to compute periodic orbits over an interval that spans the hyperbolic plateau. The escape rates computed from a few thousand periodic orbits agree with escape rates computed from Monte Carlo simulations requiring hundreds of billions of orbits.

ETLBO based optimal targeting to the moon in the PCR3BP chaotic system

The chaotic transport in Earth-Moon three-body system has been demonstrated to be a novel approach to explore the Moon at a low energy cost. However, existing targeting methods require sufficient experience to construct Earth-Moon chaotic transfer orbits, which is not an easy task to the untrained eye. In this paper, the elitist teaching-learning-based optimization (ETLBO) based optimal targeting method is presented to provide a systematic approach to design the chaotic transfer orbits in the Earth-Moon planar circular restricted three-body problem, without any requirement of prior experience. Unlike the existing targeting methods, the chaotic transfer orbits design problem is treated as a class of multi-constraints fuel-optimal problem with multi-dimensional decision variables. A discrete chaotic dynamical model is formulated according to the Poincaré map, and several consecutive control steps of small bounded thrusts are made to direct the chaotic series towards the desired invariant torus near the Moon. The suboptimal consecutive control thrusts are obtained by a state-of-art numerical optimization algorithm ETLBO, which does not require any algorithm-specific parameters with less computational effort. Numerical demonstrates are provided to illustrate the applications of the ETLBO based optimal targeting method, which reveal that several potential chaotic transfer orbits can be easily obtained by this method.

From thermonuclear fusion to Hamiltonian chaos

This paper aims at a historical and pedagogical presentation of some important contributions of the research on thermonuclear fusion by magnetic confinement to the study of Hamiltonian chaos. This chaos is defined with the help of Poincare maps on a simple two-wave Hamiltonian system. A simple criterion for computing the transition to large scale chaos is introduced. A renormalization group approach for barriers in phase space is described pictorially. The geometrical structure underlying chaos is introduced, and then described in the adiabatic limit of Hamiltonian chaos. The issue of chaotic transport is discussed in simple limit cases.