Chaotic scattering in the presence of an external magnetic field

Transport of chaotic trajectories from regions distant from or near to structures of regular motion of the Fermi-Ulam model

The chaotic portion of phase space of the simplified Fermi-Ulam model is studied under the context of transport of trajectories in two scenarios: (i) the trajectories are originated from a region distant from the islands of regular motion and are transported to a region located at a high portion of phase space and (ii) the trajectories are originated from chaotic regions around the islands of regular motion and are transported to other regions around islands of regular motion. The transport is investigated in terms of the observables histogram of transport and survival probability. We show that the histogram curves are scaling invariant and we organize the survival probability curves in four kinds of behavior, namely (a) transition from exponential decay to power law decay, (b) transition from exponential decay to stretched exponential decay, (c) transition from an initial fast exponential decay to a slower exponential decay, and (d) a single exponential decay. We show that, depending on choice of the regions of origin and destination, the transport process is weakly affected by the stickiness of trajectories around islands of regular motion.

Anomalous Magnetotransport in Disordered Structures: Classical Edge-State Percolation

By event-driven molecular dynamics simulations we investigate magnetotransport in a two-dimensional model with randomly distributed scatterers close to the field-induced localization transition. This transition is generated by percolating skipping orbits along the edges of obstacle clusters. The dynamic exponents differ significantly from those of the conventional transport problem on percolating systems, thus establishing a new dynamic universality class. This difference is tentatively attributed to a weak-link scenario, which emerges naturally due to barely overlapping edge trajectories. We make predictions for the frequency-dependent conductivity and discuss implications for active colloidal circle swimmers in a hetegogeneous environment.

Violation of adiabaticity in magnetic billiards due to separatrix crossings

We consider dynamics of magnetic billiards with curved boundaries and strong inhomogeneous magnetic field. We investigate a violation of adiabaticity of charged particle motion in this system. The destruction of the adiabatic invariance is due to the change of type of the particle trajectory: particles can drift along the boundary reflecting from it or rotate around the magnetic field at some distance from the boundary without collisions with it. Trajectories of these two types are demarcated in the phase space by a separatrix. Crossings of the separatrix result in jumps of the adiabatic invariant. We derive an asymptotic formula for such a jump and demonstrate that an accumulation of these jumps leads to the destruction of the adiabatic invariance.

Quantum chaotic scattering in graphene systems in the absence of invariant classical dynamics

Quantum chaotic scattering is referred to as the study of quantum behaviors of open Hamiltonian systems that exhibit transient chaos in the classical limit. Traditionally a central issue in this field is how the elements of the scattering matrix or their functions fluctuate as a system parameter, e.g., the electron Fermi energy, is changed. A tacit hypothesis underlying previous works was that the underlying classical phase-space structure remains invariant as the parameter varies, so semiclassical theory can be used to explain various phenomena in quantum chaotic scattering. There are, however, experimental situations where the corresponding classical chaotic dynamics can change characteristically with some physical parameter. Multiple-terminal quantum dots are one such example where, when a magnetic field is present, the classical chaotic-scattering dynamics can change between being nonhyperbolic and being hyperbolic as the Fermi energy is changed continuously. For such systems semiclassical theory is inadequate to account for the characteristics of conductance fluctuations with the Fermi energy. To develop a general framework for quantum chaotic scattering associated with variable classical dynamics, we use multi-terminal graphene quantum-dot systems as a prototypical model. We find that significant conductance fluctuations occur with the Fermi energy even for fixed magnetic field strength, and the characteristics of the fluctuation patterns depend on the energy. We propose and validate that the statistical behaviors of the conductance-fluctuation patterns can be understood by the complex eigenvalue spectrum of the generalized, complex Hamiltonian of the system which includes self-energies resulted from the interactions between the device and the semi-infinite leads. As the Fermi energy is increased, complex eigenvalues with extremely smaller imaginary parts emerge, leading to sharp resonances in the conductance.

Time irreversible billiards with piecewise-straight trajectories

We introduce a new family of billiards which break time reversal symmetry in spite of having piece-wise straight trajectories. We show that our billiards preserve the ergodic and mixing properties of conventional billiards while they may turn into exponential the power law decay of correlations characteristic of Sinai type billiards. Such billiards can be implemented by squeezing the transverse magnetic field along lines or along one-dimensional manifolds.

Stable regimes for hard disks in a channel with twisting walls

We study a gas of N hard disks in a box with semi-periodic boundary conditions. The unperturbed gas is hyperbolic and ergodic (these facts are proved for N=2 and expected to be true for all N≥2). We study various perturbations by twisting the outgoing velocity at collisions with the walls. We show that the dynamics tends to collapse to various stable regimes, however we define the perturbations, and however small they are.

Scattering off an oscillating target: Basic mechanisms and their impact on cross sections

We investigate classical scattering off a harmonically oscillating target in two spatial dimensions. The shape of the scatterer is assumed to have a boundary which is locally convex at any point and does not support the presence of any periodic orbits in the corresponding dynamics. As a simple example we consider the scattering of a beam of noninteracting particles off a circular hard scatterer. The performed analysis is focused on experimentally accessible quantities, characterizing the system, like the differential cross sections in the outgoing angle and velocity. Despite the absence of periodic orbits and their manifolds in the dynamics, we show that the cross sections acquire rich and multiple structure when the velocity of the particles in the beam becomes of the same order of magnitude as the maximum velocity of the oscillating target. The underlying dynamical pattern is uniquely determined by the phase of the first collision between the beam particles and the scatterer and possesses a universal profile, dictated by the manifolds of the parabolic orbits, which can be understood both qualitatively as well as quantitatively in terms of scattering off a hard wall. We discuss also the inverse problem concerning the possibility to extract properties of the oscillating target from the differential cross sections.

Chaotic scattering by steep repelling potentials

Consider a classical two-dimensional scattering problem: a ray is scattered by a potential composed of several tall, repelling, steep mountains of arbitrary shape. We study when the traditional approximation of this nonlinear far-from-integrable problem by the corresponding simpler billiard problem, of scattering by hard-wall obstacles of similar shape, is justified. For one class of chaotic scatterers, named here regular Sinai scatterers, the scattering properties of the smooth system indeed limit to those of the billiards. For another class, the singular Sinai scatterers, these two scattering problems have essential differences: though the invariant set of such singular scatterers is hyperbolic (possibly with singularities), that of the smooth flow may have stable periodic orbits, even when the potential is arbitrarily steep. It follows that the fractal dimension of the scattering function of the smooth flow may be significantly altered by changing the ratio between the steepness parameter and a parameter which measures the billiards' deviation from a singular scatterer. Thus, even in this singular case, the billiard scattering problem is utilized as a skeleton for studying the properties of the smooth flow. Finally, we see that corners have nontrivial and significant impact on the scattering functions.

Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space

We investigate the dynamics of chaotic trajectories in simple yet physically important Hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures. We show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one-parameter families of marginally unstable periodic orbits and is characterized by an exponent gamma=2 for the asymptotic power-law decay of the distribution of recurrence times. Generic perturbations lead to systems with hierarchical phase space, where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and Cantori. In this case, we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws, depending on the relative contribution of the primary and secondary structures of the hierarchy. Numerical verification of our main results are provided for area-preserving maps, mushroom billiards, and the newly defined magnetic mushroom billiards.

Effective dynamics in Hamiltonian systems with mixed phase space

An adequate characterization of the dynamics of Hamiltonian systems at physically relevant scales has been largely lacking. Here we investigate this fundamental problem and we show that the finite-scale Hamiltonian dynamics is governed by effective dynamical invariants, which are significantly different from the dynamical invariants that describe the asymptotic Hamiltonian dynamics. The effective invariants depend both on the scale of resolution and the region of the phase space under consideration, and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of the Hamiltonian system is modeled as a chain of hyperbolic systems.

Nonhyperbolic escape and changes in phase-space stability structures in laser-induced multiphoton dissociation of a diatomic molecule

The dependence of photodissociation of a diatomic molecule (vibrating according to the Morse potential) on the frequency of the laser field that induces it, is studied in the context of classical nonlinear dynamics. First, it is observed that as the laser frequency increases towards the harmonic frequency of the potential, a transition from stabilization due to Kolmogorov-Arnold-Moser tori to stabilization caused by the resonance stability island occurs. Then, considering the photodissociation as a nonhyperbolic half-scattering process, we investigate the influence of these changes in the phase space stability structures on dissociation dynamics via the examination of the fractal set of singularities appearing in the time-delay function of the initial state. It is found that the effective fractal dimension of this set (a finite-scale approximation of the exact dimension which is always equal to 1) and the percentage of its singularities provide a link between these changes and the dissociation rate.

Stagger-and-step method: detecting and computing chaotic saddles in higher dimensions

Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices. These transients are caused by the presence of chaotic saddles, and they are a common phenomenon in higher dimensional dynamical systems. For many physical systems, chaotic saddles have a big impact on laboratory measurements, but there has been no way to observe these chaotic saddles directly. We present the first general method to locate and visualize chaotic saddles in higher dimensions.

Equilibrium and stationary nonequilibrium states in a chain of colliding harmonic oscillators

Equilibrium and nonequilibrium properties of a chain of colliding harmonic oscillators (ding-dong model) are investigated. Our chain is modeled as harmonically bounded particles that can only interact with neighboring particles by hard-core interaction. Between the collisions, particles are just independent harmonic oscillators. We are especially interested in the stationary nonequilibrium state of the ding-dong model coupled with two stochastic heat reservoirs (not thermostated) at the ends, whose temperature is different. We check the Gallavotti-Cohen fluctuation theorem [G. Gallavoti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995)] and also the Evans-Searles identity [D. Evans and D. Searles, Phys. Rev. E. 50, 1994 (1994)] numerically. It is verified that the former theorem is satisfied for this system, although the system is not a thermostated system.

Quantum chaotic scattering with a mixed phase space: the three-disk billiard in a magnetic field

We study the classical and semiclassical scattering behavior of electrons in an open three-disk billard in the presence of a homogeneous magnetic field, which is confined to the inner part of the scattering region. As the magnetic field is increased the phase space of the invariant set of the classical scattering trajectories changes from hyperbolic (fully chaotic) to a mixed situation, where KAM tori are present. The "stickiness" of the stable trajectories leads to a much slower decay of the survival probability of trajectories as compared to the hyperbolic case. We show that this effect influences strongly the quantum fluctuations of the scattering amplitude and cross sections.