Nonequilibrium Gas and Generalized Billiards

Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.     

Negative Magneto-Resistance Beyond Weak Localization in Three-Dimensional Billiards:Effect of Arnold Diffusion

We investigate a semiclassical conductance for ballistic open three-dimensional (3-d) billiards. For partially or completely broken-ergodic 3-d billiards such as SO(2) symmetric billiards, the dependence of the conductance on the Fermi wavenumber is dramatically changed by the lead orientation. Application of a symmetry-breaking weak magnetic field brings about mixed phase-space structures of 3-d billiards which ensures a novel Arnold diffusion that cannot be seen in 2-d billiards. In contrast to the 2-d case, the anomalous increment of the conductance should inevitably include a contribution arising from Arnold diffusion as well as a weak localization correction. Discussions are devoted to the physical condition for observing this phenomenon.     

Nonergodicity of the motion in three-dimensional steep repelling dispersing potentials

It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems that are arbitrarily close to three-dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are nonergodic. The mechanism for creating the islands is corners of the billiards domain.     

Chaos in composite billiards

The mechanisms and features of the chaotic behavior in billiards with ray splitting (refraction) are considered. In contrast to ordinary billiards, the law of motion in composite billiards that is coded with a sequence of ray visits to different media is shown to be deterministically chaotic. The analysis is performed in terms of a geometrical-dynamical approach in which a symmetric phase space is used instead of the ordinary Hamiltonian phase space. The chaotization elements in composite billiards of a general position are studied. The dynamics of rays in ring billiards consisting of two concentric media with different refractive indices is considered.     

Rashba billiards

We study the energy levels of non-interacting electrons confined to move in two-dimensional billiard regions and having a spin-dependent dynamics due to a finite Rashba spin splitting. The free space Green's function for such Rashba billiards is constructed analytically and used to find the area and perimeter contributions to the density of states, as well as the corresponding smooth counting function. We show that, in contrast to systems with spin-rotational invariance, Rashba billiards always possess a negative energy spectrum. A semi-classical analysis is presented to interpret the singular behavior of the density of states at certain negative energies for circular Rashba billiards. Our detailed analysis of the spin structure of circular Rashba billiards reveals a finite out-of-plane spin projection for electron eigenstates.     

Track Billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of 3-dimensional billiards. Interestingly, we find that for some track billiards, the mechanism generating hyperbolicity is not the defocusing one, which requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary.     

Numerical exploration of a family of strictly convex billiards with boundary of classC 2

We are interested in the possible existence of strictly convex ergodic billiards. Such billiards are searched for by means of numerical investigation. The boundary of a billiard is built with four arcs of classC . Adjacent arcs have equal curvatures at connecting points. The surface of section of the billiards is explored. It seems as if symmetric billiards always have invariant curves (islands). Asymmetric billiards have been found which look ergodic. They are built with an arc of an ellipse, two arcs of circles, and one-half of a Descartes oval.     

Local instability of orbits in polygonal and polyhedral billiards

We classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons, for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.     

Scars of periodic orbits in the stadium action billiard

Compact billiards in phase space, or action billiards, are constructed by truncating the classical Hamiltonian in the action variables. The corresponding quantum mechanical system has a finite Hamiltonian matrix. In previous papers we defined the compact analog of common billiards, i.e., straight motion in phase space followed by specular reflections at the boundaries. Computation of their quantum energy spectra establishes that their properties are exactly those of common billiards: the short-range statistics follow the known universality classes depending on the regular or chaotic nature of the motion, while the long-range fluctuations are determined by the periodic orbits. In this work we show that the eigenfunctions also follow qualitatively the general characteristics of common billiards. In particular, we show that the low-lying levels can be classified according to their nodal lines as usual and that the high excited states present scars of several short periodic orbits. Moreover, since all the eigenstates of action billiards can be computed with great accuracy, Bogomolny's semiclassical formula for the scars can also be tested successfully.     

Semi-Focusing Billiards: Hyperbolicity

In this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of ℝ³. We also prove that a related class of semi-focusing billiards has mixed dynamics, i.e., their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other. These billiards are the first rigorous examples of billiards in domains of ℝ³ with divided phase space. The first author was partially supported by the NSF grant #0140165 and the Humboldt Foundation. The second author was partially supported by the FCT (Portugal) through the Program POCTI/FEDER.     

正方体量子台球的半经典分析(英文)

近几年,量子台球问题引起人们的广泛兴趣.以前有很多对二维量子台球做过研究,相对于二维台球来说,三维台球更接近实际体系.本文以三维正方体量子台球为例,利用半经典闭合轨道理论计算了正方体量子台球中的经典开轨道,并研究量子谱函数与经典轨道长度之间的对应关系,发现他们之间对应的很好.这将有助于我们分析开放型量子台球中输运性质问题.利用这种方法物理图像清晰,计算量小并且可以帮助理解一些混沌体系的性质.这是半经典理论为联系量子力学与经典力学起桥梁作用的又一证明.     

Elliptic Flowers: New Types of Dynamics to Study Classical and Quantum Chaos

We construct examples of billiards where two chaotic flows are moving in opposite directions around a non-chaotic core or vice versa; the dynamics in the core are chaotic but flows that are moving in opposite directions around it are non-chaotic. These examples belong to a new class of dynamical systems called elliptic flowers billiards. Such systems demonstrate a variety of new behaviors which have never been observed or predicted to exist. Elliptic flowers billiards, where a chaotic (non-chaotic) core coexists with the same (chaotic/non-chaotic) type of dynamics in flows were recently constructed. Therefore, all four possible types of coexisting dynamics in the core and tracks are detected. However, it is just the beginning of studies of elliptic flowers billiards, which have already extended the imagination of what may happen in phase spaces of nonlinear systems. We outline some further directions of investigation of elliptic flowers billiards, which may bring new insights into our understanding of classical and quantum dynamics in nonlinear systems.     

New mechanism of chaos in triangular billiards

A new mechanism of weak chaos in triangular billiards has been proposed owing to the effect of cutting of beams of rays. A similar mechanism is also implemented in other polygonal billiards. Cutting of beams results in the separation of initially close rays at a finite angle by jumps in the process of reflections of beams at the vertices of a billiard. The opposite effect of joining of beams of rays occurs in any triangular billiard along with cutting. It has been shown that the cutting of beams has an absolute character and is independent of the form of a triangular billiard or the parameters of a beam. On the contrary, joining has a relative character and depends on the commensurability of the angles of the triangle with π. Joining always suppresses cutting in triangular billiards whose angles are commensurable with π. For this reason, their dynamics cannot be chaotic. In triangular billiards whose angles are rationally incommensurable with π, cutting always dominates, leading to weak chaos. The revealed properties are confirmed by numerical experiments on the phase portraits of typical triangular billiards.     

The influence of confining wall profile on quantum interference effects in etched Ga0.25In0.75As/InP billiards

We present measurements of the potential profile of etched GaInAs/InP billiards and show that their energy gradients are an order of magnitude steeper than those of surface-gated GaAs/AlGaAs billiards. Previously observed in GaAs/AlGaAs billiards, fractal conductance fluctuations are predicted to be critically sensitive to the billiard profile. Here we show that, despite the increase in energy gradient, the fractal conductance fluctuations persist in the harder GaInAs/InP billiards.     

The K-property of “orthogonal” cylindric billiards

Toric billiards with cylindric scatterers (briefly cylindric billards) generalize the class of Hamiltonian systems of elastic hard balls. In this paper a class of cylindric billiards is considered where the cylinders are orthogonal or more exactly: the constituent space of any cylindric scatterer is spanned by some of the (of course, orthogonal) coordinate vectors adapted to the euclidean torus. It is shown that the natural necessary condition for the K-property of such billiards is also sufficient.Research supported by the Hungarian National Foundation for Scientific Research, grant No. 1902     

Properties of Some Chaotic Billiards with Time-Dependent Boundaries

A dispersing billiard (Lorentz gas) and focusing billiards (in the form of a stadium) with time-dependent boundaries are considered. The problem of a particle acceleration in such billiards is studied. For the Lorentz gas two cases of the time-dependence are investigated: stochastic perturbations of the boundary and its periodic oscillations. Two types of focusing billiards with periodically forced boundaries are explored: stadium with strong chaotic properties and a near-rectangle stadium. It is shown that in all cases billiard particles can reach unbounded velocities. Average velocities of the particle ensemble as functions of time and the number of collisions are obtained.     

Entanglement distribution statistic in Andreev billiards

We investigate statistical aspects of the entanglement production for open chaotic mesoscopic billiards in contact with superconducting parts, known as Andreev billiards. The complete distributions of concurrence and entanglement of formation are obtained by using the Altland–Zirnbauer symmetry classes of circular ensembles of scattering matrices, which complements previous studies in chaotic universal billiards belonging to other classes of random matrix theory. Our results show a unique and very peculiar behavior: the realization of entanglement in a Andreev billiard always results in non-separable state, regardless of the time reversal symmetry. The analytical calculations are supported by a numerical Monte Carlo simulation.     

Mixing of wavefunctions in rectangular microwave billiards

A set-up is described allowing the automatic registration of wavefunctions of quasi-two-dimensional microwave billiards of arbitrary shape. Tests of the apparatus with rectangular shaped billiards showed that a precision of some percent in the wavefunction amplitudes can be obtained, as far as isolated resonances are considered. For the case of overlapping resonances, however, the measurement yields wavefunctions which are close to a symmetric and an antisymmetric linear combination of the original rectangle eigenfunctions. The cause for this at first sight surprising result is discussed. Received 21 January 2000     

Classical projected phase space density of billiards and its relation to the quantum neumann spectrum

A comparison of classical and quantum evolution usually involves a quasiprobability distribution as a quantum analogue of the classical phase space distribution. In an alternate approach that we adopt here, the classical density is projected on to the configuration space. We show that for billiards, the eigenfunctions of the coarse-grained projected classical evolution operator are identical to a first approximation to the quantum Neumann eigenfunctions. However, even though there exists a correspondence between the respective eigenvalues, their time evolutions differ. This is demonstrated numerically for the stadium and lemon-shaped billiards.     

variant Planck's over 2pi expansion for the periodic orbit quantization of chaotic systems

We report the results of a periodic orbit quantization of classically chaotic billiards beyond Gutzwiller approximation in terms of asymptotic series in powers of the Planck constant (or in powers of the inverse of the wave number kappa in billiards). We derive explicit formulas for the kappa(-1) approximation of our semiclassical expansion. We illustrate our theory with the classically chaotic scattering of a wave on three disks. The accuracy on the real parts of the scattering resonances is improved by one order of magnitude.