Open Billiards Research Articles

Open Billiards: Invariant and Conditionally Iinvariant Probabilities on Cantor Sets

Billiards are the simplest models for understanding the statistical theory of the dynamics of a gas in a closed compartment. We analyze the dynamics of a class of billiards (the open billiard on the plane) in terms of invariant and conditionally invariant probabilities. The dynamical system has a horseshoe structure. The stable and unstable manifolds are analytically described. The natural probability $\mu $ is invariant and has support in a Cantor set. This probability is the conditional limit of a conditional probability $\mu _F $ that has a density with respect to the Lebesgue measure. A formula relating entropy, Lyapunov exponent, and Hausdorff dimension of a natural probability $\mu $ for the system is presented. The natural probability $\mu $ is a Gibbs state of a potential$\psi $ (cohomologous to the potential associated to the positive Lyapunov exponent; see formula (0.1)), and we show that for a dense set of such billiards the potential tb is not lattice. As the system has a horseshoe structure, one can compute the asymptotic growth rate of $n( r )$, the number of closed trajectories with the largest eigenvalue of the derivative smaller than r. This theorem implies good properties for the poles of the associated Zeta function and this result turns out to be very important for the understanding of scattering quantum billiards.

Quantum transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.

Quantum conductance fluctuations and classical short-path dynamics.

We present numerical results for ballistic-electron quantum transport through weakly open integrable circle and chaotic stadium billiards. The geometry of the pair of conducting leads is chosen in accordance with recent experiments for semiconductor microstructures [Marcus [ital et] [ital al]., Phys. Rev. Lett. [bold 69], 506 (1992)]. The conductance as a function of the Fermi wave number displays characteristic noisy fluctuations for both the integrable and the chaotic systems. We show that structures in the conductance autocorrelation function as a function of the Fermi wave number are related to short-length classical orbits. This correspondence permits incorporation of effects of phase decoherence due to incoherent scattering into the quantum calculation.

Magnetoconductance in Open Stadium Billiard: Quantum Analogue of Transition from Chaos to Tori

Quantum and classical analyses are made on transport in an open Bunimovich stadium billiard in the perpendicular magnetic field B . The mild and violent undulations of quantum conductance g ( B ) in low- and high-field regions are due to the stability of the fully chaotic phase space in the underlying classical dynamics and to the transitional instability of the ergodic part which is rapidly replaced by the Kolmogorov-Arnold-Moser tori, respectively. This crossover is also signaled by a marked difference between gradients of cusp-shaped central peaks in the autocorrelation functions.

Magneto-conductance in open billiards: comparison between circle and stadium

A comparative study is presented on quantum transports in weakly opened circle and stadium billiards in the perpendicular magnetic field B. While in the circle the magnetoconductance g(B) shows grossly regular oscillations, in the stadium it exhibits a transition from mild to violent undulations with increase of B. The rich fluctuation features of g(B), characterized by gradients of the cusp-like central peaks in autocorrelation functions, are attributed to the stability and instability of phase space in the underlying classical dynamics.

Quantum Transport in Open Billiards: Comparison between Circle and Stadium

Quantum transport is studied for weakly-opened integrable circle (C) and fully chaotic stadium (S) billiards with a pair of conducting leads. Electric conductance vs Fermi wavenumber for each incident mode, when smoothed, exhibits short-period and small-amplitude oscillations around some plateau value for C billiard, whereas long-period and large-amplitude smooth oscillations for S billiard. The wavenumber correlation for the conductance is much stronger in S than C billiards. These contrasts can be understood in terms of regular and chaotic orbits in the underlying classical dynamics.

Quantum Transport in Open Billiards: Dependence on Degree of Opening

Quantum transport is studied for open billiards with a pair of conducting leads. Boundaries of the billiards are of squared-cosine shape and the electric conductance as a function of the Fermi wave number is computed by tuning the degree of opening. A quantized plateau accompanied by a sequence of intermittent sharp dips in a `strongly opened' case changes into anomalously noisy fluctuations in a `weakly opened' case. In the latter case, however, the smoothed conductance shows regular oscillations around some plateau values. The failure of the adiabatic approximation is also discussed.

Decay of a chaotic dynamical system

Decay of chaotic open stadium billiards is examined numerically. Certain open stadium systems are shown to exhibit other than simple exponential decay (i.e. nonstatistical) behavior. In addition, algebraic long time decay occurs in some cases. Observed decays are modeled with a minimally dynamic hybrid statistical theory based on mixing assumptions. These results are discussed in terms of other related results and those associated with integrable systems.

The Klein paradox and superradiance: Corinne A. Manogue. Center for Relativity, University of Texas, Austin, Texas 78712

We investigate the probability distributions and spatial correlations of the local densities of electron wave functions for ballistic transport through mesoscopic chaotic open billiards. By quantitative comparison between our accurate fully-quantal calculations with theoretical expressions, we find wave-statistical behaviors intrinsically different from those in closed systems. It is shown that chaotic-scattering wave functions in open systems can be universally interpreted in terms of statistically independent real and imaginary random fields together with breaking of the endowed space reciprocity, resulting in the same wave function statistics as in the time-reversal symmetry-breaking crossover regime in closed systems.