Exponential Divergence and Long Time Relaxation in Chaotic Quantum Dynamics

Phase space representations of the dynamics of the quantal and classical cat map are used to explore quantum--classical correspondence in a K-system: as $\hbar \to 0$, the classical chaotic behavior is shown to emerge smoothly and exactly. The quantum dynamics near the classical limit displays both exponential separation of adjacent distributions and long time relaxation, two characteristic features of classical chaotic motion.Comment: 10 pages, ReVTeX, to appear in Phys. Rev. Lett. 13 figures NOT included. Available either as LARGE (uuencoded gzipped) postscript files or hard-copies from [email protected]

Instability of the massive Klein-Gordon field on the Kerr spacetime

We investigate the instability of the massive scalar field in the vicinity of a rotating black hole. The instability arises from amplification caused by the classical superradiance effect. The instability affects bound states: solutions to the massive Klein-Gordon equation which tend to zero at infinity. We calculate the spectrum of bound state frequencies on the Kerr background using a continued fraction method, adapted from studies of quasinormal modes. We demonstrate that the instability is most significant for the $l = 1$, $m = 1$ state, for $M \mu \lesssim 0.5$. For a fast rotating hole ($a = 0.99$) we find a maximum growth rate of $\tau^{-1} \approx 1.5 \times 10^{-7} (GM/c^3)^{-1}$, at $M \mu \approx 0.42$. The physical implications are discussed.Comment: Added references. 27 pages, 7 figure

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.

The XXL Survey: XII. Optical spectroscopy of X-ray-selected clusters and the frequency of AGN in superclusters

This article belongs to the first series of XXL publications. It presents multifibre spectroscopic observations of three 0.55 sq.deg. fields in the XXL Survey, which were selected on the basis of their high density of X-ray-detected clusters. The observations were obtained with the AutoFib2+WYFFOS (AF2) wide-field fibre spectrograph mounted on the 4.2m William Herschel Telescope. The paper first describes the scientific rationale, the preparation, the data reduction, and the results of the observations, and then presents a study of active galactic nuclei (AGN) within three superclusters. We obtained redshifts for 455 galaxies in total, 56 of which are counterparts of X-ray point-like sources. We were able to determine the redshift of the merging supercluster XLSSC-e, which consists of six individual clusters at z~0.43, and we confirmed the redshift of supercluster XLSSC-d at z~0.3. More importantly, we discovered a new supercluster, XLSSC-f, that comprises three galaxy clusters also at z~0.3. We find a significant 2D overdensity of X-ray point-like sources only around the supercluster XLSSC-f. This result is also supported by the spatial (3D) analysis of XLSSC-f, where we find four AGN with compatible spectroscopic redshifts and possibly one more with compatible photometric redshift. In addition, we find two AGN (3D analysis) at the redshift of XLSSC-e, but no AGN in XLSSC-d. Comparing these findings with the optical galaxy overdensity we conclude that the total number of AGN in the area of the three superclusters significantly exceeds the field expectations. The difference in the AGN frequency between the three superclusters cannot be explained by the present study because of small number statistics. Further analysis of a larger number of superclusters within the 50 sq. deg. of the XXL is needed before any conclusions on the effect of the supercluster environment on AGN can be reached.Comment: 11 pages, published by A&

Galaxy pre-processing in substructures around z∼\sim0.4 galaxy clusters

We present a detailed analysis of galaxy colours in two galaxy clusters at \mbox{z $\sim$ 0.4}, \mbox{MACS J0416.1-2403} and \mbox{MACS J1206.2-0847}, drawn from the CLASH-VLT survey, to investigate the role of pre-processing in the quenching of star formation. We estimate the fractions of red and blue galaxies within the main cluster and the detected substructures and study the trends of the colour fractions as a function of the projected distance from the cluster and substructure centres. Our results show that the colours of cluster and substructure members have consistent spatial distributions. In particular, the colour fractions of galaxies inside substructures follow the same spatial trends observed in the main clusters. Additionally, we find that at large cluster-centric distances \mbox{($r \geq r_{200}$)} the fraction of blue galaxies in both the main clusters and in the substructures is always lower than the average fraction of UVJ-selected star-forming galaxies in the field as measured in the COSMOS/UltraVista data set. We finally estimate environmental quenching efficiencies in the clusters and in the substructures and find that at large distances from the cluster centres, the quenching efficiency of substructures becomes comparable to the quenching efficiency of clusters. Our results suggest that pre-processing plays a significant role in the formation and evolution of passive galaxies in clusters at low redshifts.Comment: Accepted for publication in MNRAS. 28 pages, 14 figures, 20 table

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

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

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

GASP - XVII. H I imaging of the jellyfish galaxy JO206:Gas stripping and enhanced star formation

We present VLA HI observations of JO206, a prototypical ram-pressure stripped galaxy in the GASP sample. This massive galaxy (M$_{\ast} =$ 8.5 $\times$ 10$^{10}$ M$_{\odot}$) is located at a redshift of $z =$ 0.0513, near the centre of the low-mass galaxy cluster, IIZw108 ($\sigma \sim575$ km/s). JO206 is characterised by a long tail ($\geq$90 kpc) of ionised gas stripped away by ram-pressure. We find a similarly long HI tail in the same direction as the ionised gas tail and measure a total HI mass of $3.2 \times 10^{9}$ M$_{\odot}$. This is about half the expected HI mass given the stellar mass and surface density of JO206. A total of $1.8 \times 10^{9}$ M$_{\odot}$ (60%) of the detected HI is in the gas stripped tail. An analysis of the star formation rate shows that the galaxy is forming more stars compared to galaxies with the same stellar and HI mass. On average we find a HI gas depletion time of $\sim$0.5 Gyr which is about four times shorter than that of "normal" spiral galaxies. We performed a spatially resolved analysis of the relation between star formation rate density and gas density in the disc and tail of the galaxy at the resolution of our HI data. The star formation efficiency of the disc is about 10 times higher than that of the tail at fixed HI surface densities. Both the inner and outer parts of JO206 show an enhanced star formation compared to regions of similar HI surface density in field galaxies. The enhanced star formation is due to ram-pressure stripping during the galaxy's first infall into the cluster.Comment: 13 pages, 12 figures, Accepted for publication in MNRA