Geometrical Models of the Phase Space Structures Governing Reaction Dynamics

Hamiltonian dynamical systems possessing equilibria of ${saddle} \times {centre} \times...\times {centre}$ stability type display \emph{reaction-type dynamics} for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow \emph{bottlenecks} created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a \emph{Normally Hyperbolic Invariant Manifold (NHIM)}, whose \emph{stable and unstable manifolds} have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behavior. This NHIM forms the natural (dynamical) equator of a (spherical) \emph{dividing surface} which locally divides an energy surface into two components (`reactants' and `products'), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in \emph{transition state theory} where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface. We discuss three presentations of the energy surface and the phase space structures contained in it for 2-degree-of-freedom (DoF) systems in the threedimensional space $\R^3$, and two schematic models which capture many of the essential features of the dynamics for $n$-DoF systems. In addition, we elucidate the structure of the NHIM.Comment: 44 pages, 38 figures, PDFLaTe

A detailed study of quasinormal frequencies of the Kerr black hole

We compute the quasinormal frequencies of the Kerr black hole using a continued fraction method. The continued fraction method first proposed by Leaver is still the only known method stable and accurate for the numerical determination of the Kerr quasinormal frequencies. We numerically obtain not only the slowly but also the rapidly damped quasinormal frequencies and analyze the peculiar behavior of these frequencies at the Kerr limit. We also calculate the algebraically special frequency first identified by Chandrasekhar and confirm that it coincide with the $n=8$ quasinormal frequency only at the Schwarzschild limit.Comment: REVTEX, 15 pages, 7 eps figure

Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape

Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems. While the former approach studies how regions of phase space are transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing periodic orbit around saddles. Both of these frameworks require computation with curves represented by millions of points-computing intersection points between these curves and area bounded by the segments of these curves-for quantifying the transport and escape rate. We present a theory for computing these intersection points and the area bounded between the segments of these curves based on a classification of the intersection points using equivalence class. We also present an alternate theory for curves with nontransverse intersections and a method to increase the density of points on the curves for locating the intersection points accurately.The numerical implementation of the theory presented herein is available as an open source software called Lober. We used this package to demonstrate the application of the theory to lobe dynamics that arises in fluid mechanics, and rate of escape from a potential well that arises in ship dynamics.Comment: 33 pages, 17 figure

Unconventional Gravitational Excitation of a Schwarzschild Black Hole

Besides the well-known quasinormal modes, the gravitational spectrum of a Schwarzschild black hole also has a continuum part on the negative imaginary frequency axis. The latter is studied numerically for quadrupole waves. The results show unexpected striking behavior near the algebraically special frequency $\Omega=-4i$. This reveals a pair of unconventional damped modes very near $\Omega$, confirmed analytically.Comment: REVTeX4, 4pp, 6 EPS figure files. N.B.: "Alec" is my first, and "Maassen van den Brink" my family name. v2: better pole placement in Fig. 1. v3: fixed Refs. [9,20]. v4: added context on "area quantum" research; trimmed one Fig.; textual clarification

From Heisenberg matrix mechanics to EBK quantization: theory and first applications

Despite the seminal connection between classical multiply-periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical (EBK) quantization of bound states, we show that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of action variables. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heisenberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semiclassical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest several new or modified applications of EBK quantization.Comment: 37 pages including 3 poscript figures, submitted to Phys. Rev.

Early Science with the Large Millimeter Telescope: COOL BUDHIES I - a pilot study of molecular and atomic gas at z~0.2

An understanding of the mass build-up in galaxies over time necessitates tracing the evolution of cold gas (molecular and atomic) in galaxies. To that end, we have conducted a pilot study called CO Observations with the LMT of the Blind Ultra-Deep H I Environment Survey (COOL BUDHIES). We have observed 23 galaxies in and around the two clusters Abell 2192 (z = 0.188) and Abell 963 (z = 0.206), where 12 are cluster members and 11 are slightly in the foreground or background, using about 28 total hours on the Redshift Search Receiver (RSR) on the Large Millimeter Telescope (LMT) to measure the $^{12}$CO J = 1 --> 0 emission line and obtain molecular gas masses. These new observations provide a unique opportunity to probe both the molecular and atomic components of galaxies as a function of environment beyond the local Universe. For our sample of 23 galaxies, nine have reliable detections (S/N$\geq$3.6) of the $^{12}$CO line, and another six have marginal detections (2.0 < S/N < 3.6). For the remaining eight targets we can place upper limits on molecular gas masses roughly between $10^9$ and $10^{10} M_\odot$. Comparing our results to other studies of molecular gas, we find that our sample is significantly more abundant in molecular gas overall, when compared to the stellar and the atomic gas component, and our median molecular gas fraction lies about $1\sigma$ above the upper limits of proposed redshift evolution in earlier studies. We discuss possible reasons for this discrepancy, with the most likely conclusion being target selection and Eddington bias.Comment: MNRAS, submitte

Non-Hermitian matrix description of the PT symmetric anharmonic oscillators

Schroedinger equation H \psi=E \psi with PT - symmetric differential operator H=H(x) = p^2 + a x^4 + i \beta x^3 +c x^2+i \delta x = H^*(-x) on L_2(-\infty,\infty) is re-arranged as a linear algebraic diagonalization at a>0. The proof of this non-variational construction is given. Our Taylor series form of \psi complements and completes the recent terminating solutions as obtained for certain couplings \delta at the less common negative a.Comment: 18 pages, latex, no figures, thoroughly revised (incl. title), J. Phys. A: Math. Gen., to appea

Diffuse-Charge Dynamics in Electrochemical Systems

The response of a model micro-electrochemical system to a time-dependent applied voltage is analyzed. The article begins with a fresh historical review including electrochemistry, colloidal science, and microfluidics. The model problem consists of a symmetric binary electrolyte between parallel-plate, blocking electrodes which suddenly apply a voltage. Compact Stern layers on the electrodes are also taken into account. The Nernst-Planck-Poisson equations are first linearized and solved by Laplace transforms for small voltages, and numerical solutions are obtained for large voltages. The ``weakly nonlinear'' limit of thin double layers is then analyzed by matched asymptotic expansions in the small parameter $\epsilon = \lambda_D/L$, where $\lambda_D$ is the screening length and $L$ the electrode separation. At leading order, the system initially behaves like an RC circuit with a response time of $\lambda_D L / D$ (not $\lambda_D^2/D$), where $D$ is the ionic diffusivity, but nonlinearity violates this common picture and introduce multiple time scales. The charging process slows down, and neutral-salt adsorption by the diffuse part of the double layer couples to bulk diffusion at the time scale, $L^2/D$. In the ``strongly nonlinear'' regime (controlled by a dimensionless parameter resembling the Dukhin number), this effect produces bulk concentration gradients, and, at very large voltages, transient space charge. The article concludes with an overview of more general situations involving surface conduction, multi-component electrolytes, and Faradaic processes.Comment: 10 figs, 26 pages (double-column), 141 reference

Recent ecological change in ancient lakes

Ancient lakes are among the best archivists of past environmental change, having experienced more than one full glacial cycle, a wide range of climatic conditions, tectonic events, and long association with human settlements. These lakes not only record long histories of environmental variation and human activity in their sediments, but also harbor very high levels of biodiversity and endemism. Yet, ancient lakes are faced with a familiar suite of anthropogenic threats, which may degrade the unusual properties that make them especially valuable to science and society. In all ancient lakes for which data exist, significant warming of surface waters has occurred, with a broad range of consequences. Eutrophication threatens both native species assemblages and regional economies reliant on clean surface water, fisheries, and tourism. Where sewage contributes nutrients and heavy metals, one can anticipate the occurrence of less understood emerging contaminants, such as pharmaceuticals, personal care products, and microplastics that negatively affect lake biota and water quality. Human populations continue to increase in most of the ancient lakes’ watersheds, which will exacerbate these concerns. Further, human alterations of hydrology, including those produced through climate change, have altered lake levels. Co‐occurring with these impacts have been intentional and unintentional species introductions, altering biodiversity. Given that the distinctive character of each ancient lake is strongly linked to age, there may be few options to remediate losses of species or other ecosystem damage associated with modern ecological change, heightening the imperative for understanding these systems

Wigner's quantum phase space current in weakly anharmonic weakly excited two-state systems

This is an open access article distributed under the terms of the Creative Commons Attribution License CC BY 4.0 (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.There are no phase-space trajectories for anharmonic quantum systems, but Wigner’s phase-space representation of quantum mechanics features Wigner current J . This current reveals fine details of quantum dynamics – finer than is ordinarily thought accessible according to quantum folklore invoking Heisenberg’s uncertainty principle. Here, we focus on the simplest, most intuitive, and analytically accessible aspects of J . We investigate features of J for bound states of time-reversible, weakly-anharmonic one-dimensional quantum-mechanical systems which are weakly-excited. We establish that weakly-anharmonic potentials can be grouped into three distinct classes: hard, soft, and odd potentials. We stress connections between each other and the harmonic case. We show that their Wigner current fieldline patterns can be characterised by J ’s discrete stagnation points, how these arise and how a quantum system’s dynamics is constrained by the stagnation points’ topological charge conservation. We additionally show that quantum dynamics in phase space, in the case of vanishing Planck constant ̄ h or vanishing anharmonicity, does not pointwise converge to classical dynamics.Peer reviewe