Abstract:
Over the preceeding twenty years, the role of underlying classical dynamics in quantum mechanical tunneling has received considerable attention. A number of new tunneling phenomena have been uncovered that have been directly linked to the set of dynamical possibilities arising in simple systems that contain at least some chaotic motion. These tunneling phenomena can be identified by their novel $\hbar$-dependencies and/or statistical behaviors. We summarize a sampling of these phenomena and mention some applications.

Abstract:
Using semiclassical methods, it is possible to construct very accurate approximations in the short wavelength limit of quantum dynamics that rely exclusively on classical dynamical input. For systems whose classical realization is strongly chaotic, there is an exceedingly short logarithmic Ehrenfest time scale, beyond which the quantum and classical dynamics of a system necessarily diverge, and yet the semiclassical construction remains valid far beyond that time. This fact leads to a paradox if one ponders the reversibility and predictability properties of quantum and classical mechanics. They behave very differently relative to each other, with classical dynamics being essentially irreversible/unpredictable, whereas quantum dynamics is reversible/stable. This begs the question, "how can an accurate approximation to a reversible/stable dynamics be constructed from an irreversible/unpredictable one?" The resolution of this incongruity depends on a couple key ingredients, a well-known, inherent, one-way structural stability of chaotic systems and an overlap integral not being amenable to the saddle point method.

Abstract:
The Kolmogorov-Sinai (K-S) entropy is a central measure of complexity and chaos. Its calculation for many-body systems is an interesting and important challenge. In this paper, the evaluation is formulated by considering $N$-dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem. This approach makes explicit a duality relation that is exactly analogous to one found in a generalized Anderson tight-binding model, and leads to a formally exact expression for the finite-time K-S entropy. Within this formalism there is a hierarchy of approximations, the final one being a diagonal approximation that only makes use of instantaneous Hessians of the potential to find the K-S entropy. By way of a non-trivial illustration, the K-S entropy of $N$ identically coupled kicked rotors (standard maps) is investigated. The validity of the various approximations with kicking strength, particle number, and time are elucidated. An analytic formula for the K-S entropy within the diagonal approximation is derived and its range of validity is also explored.

Abstract:
Some statistical properties of finite-time stability exponents in the standard map can be estimated analytically. The mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants. Whereas the mean carries information about the strength of the interaction, and only indirect information about dynamical correlations, the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength. In particular, the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time-scalings. The average of the stability matrix' inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter. The usual accelerator modes and the small islands created through double saddle node bifurcations, which come halfway between the positions in interaction strength of the usual accelerator modes, are clearly visible in the variance, whose time scaling is capable of detecting the presence of islands as small as 0.01% of the phase space. We study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions. We also discuss the nature of this local approximation in some detail.

Abstract:
A general relation is derived for the action difference between two fixed points and a phase space area bounded by the irreducible component of a heteroclinic tangle. The determination of this area can require accurate calculation of heteroclinic orbits, which are important in a wide range of dynamical system problems. For very strongly chaotic systems initial deviations from a true orbit are magnified by a large exponential rate making direct computational methods fail quickly. Here, a method is developed that avoids direct calculation of the orbit by making use of the well-known stability property of the invariant unstable and stable manifolds. Under an area-preserving map, this property assures that any initial deviation from the stable (unstable) manifold collapses onto themselves under inverse (forward) iterations of the map. Using a set of judiciously chosen auxiliary points on the manifolds, long orbit segments can be calculated using the stable and unstable manifold intersections of the heteroclinic (homoclinic) tangle. Detailed calculations using the example of the kicked rotor are provided along with verification of the relation between action differences. The loop structure of the heteroclinic tangle is necessarily quite different from that of the turnstile for a homoclinic tangle, its analogous partner.

Abstract:
The sensitivity of a wave field's evolution to small perturbations is of fundamental interest. For chaotic systems, there are two distinct regimes of either exponential or Gaussian overlap decay in time. We develop a semiclassical approach for understanding both regimes, and give a simple expression for the crossover time between the regimes. The wave field's evolution is considerably more stable than the exponential instability of chaotic trajectories seem to suggest. The resolution of this paradox lies in the collective behavior of the appropriate set of trajectories. Results are given for the standard map.

Abstract:
In a recent letter [Europhys. Lett. 97, 34002 (2012)], random matrix theory is introduced for long-range acoustic propagation in the ocean. The theory is expressed in terms of unitary propagation matrices that represent the scattering between acoustic modes due to sound speed fluctuations induced by the ocean's internal waves. The scattering exhibits a power-law decay as a function of the differences in mode numbers thereby generating a power-law, banded, random unitary matrix ensemble. This work gives a more complete account of that approach and extends the methods to the construction of an ensemble of acoustic timefronts. The result is a very efficient method for studying the statistical properties of timefronts at various propagation ranges that agrees well with propagation based on the parabolic equation. It helps identify which information about the ocean environment survives in the timefronts and how to connect features of the data to the surviving environmental information. It also makes direct connections to methods used in other disordered wave guide contexts where the use of random matrix theory has a multi-decade history.

Abstract:
In quantum/wave systems with chaotic classical analogs, wavefunctions evolve in highly complex, yet deterministic ways. A slight perturbation of the system, though, will cause the evolution to diverge from its original behavior increasingly with time. This divergence can be measured by the fidelity, which is defined as the squared overlap of the two time evolved states. For chaotic systems, two main decay regimes of either Gaussian or exponential behavior have been identified depending on the strength of the perturbation. For perturbation strengths intermediate between the two regimes, the fidelity displays both forms of decay. By applying a complementary combination of random matrix and semiclassical theory, a uniform approximation can be derived that covers the full range of perturbation strengths. The time dependence is entirely fixed by the density of states and the so-called transition parameter, which can be related to the phase space volume of the system and the classical action diffusion constant, respectively. The accuracy of the approximations are illustrated with the standard map.

Abstract:
Ocean acoustic propagation can be formulated as a wave guide with a weakly random medium generating multiple scattering. Twenty years ago, this was recognized as a quantum chaos problem, and yet random matrix theory, one pillar of quantum or wave chaos studies, has never been introduced into the subject. The modes of the wave guide provide a representation for the propagation, which in the parabolic approximation is unitary. Scattering induced by the ocean's internal waves leads to a power-law random banded unitary matrix ensemble for long-range deep ocean acoustic propagation. The ensemble has similarities, but differs, from those introduced for studying the Anderson metal-insulator transition. The resulting long-range propagation ensemble statistics agree well with those of full wave propagation using the parabolic equation.

Abstract:
In a recent Phys. Rev. E Rapid Communication, the authors, Silvestrov and Beenakker, introduce a way to lengthen the Ehrenfest time, $\tau$, for fully chaotic systems. We disagree with several statements made in their paper, and address the following points essential to their conclusions: 1) it is not true that all semiclassical approximations for chaotic systems fail at a so-called `logtime', $\tau \propto - \ln (\hbar)$, differing only by a numerical coefficient; and 2) the limitation of the semiclassical approximation as expressed in the authors' Eq. (8) is not limited by their argument leading to Eq. (12).