Marko Robnik

Semiclassical level spacings when regular and chaotic orbits coexist

The authors calculate semiclassical limiting level spacing distributions P(S) for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic. In the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase-space regions, sequences from regular regions, having Poisson distributions and those from irregular regions having Wigner distributions. The formulae for P(S) depend on the sum of the Liouville measures of all the classical regular regions, and on the separate Liouville measures of the significant chaotic regions.

Classical dynamics of a family of billiards with analytic boundaries

The classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated. The author gives the stability analysis of major periodic orbits, present the Poincare maps, demonstrate the mixing properties by following the evolution of a small element in phase space, show the existence of homoclinic points, and calculate the Lyapunov exponent and the Kolmogorov entropy h. It turns out that the system becomes strongly chaotic (positive h) for sufficiently large deformations of the unit disc. The system shows a generic stochastic transition. The computations suggest that the system is mixing if the boundary is not convex.

False time-reversal violation and energy level statistics: the role of anti-unitary symmetry

The authors extend the classification of symmetries necessary to predict the universality class of spectral fluctuations of quantal systems whose classical motion is chaotic, by explaining that a system with neither time-reversal symmetry (T) nor geometric symmetry may display the spectral statistics of the Gaussian orthogonal ensemble (GOE), rather than those of the Gaussian unitary ensemble (GUE), provided it possesses instead some combination of symmetries which includes T. Such combinations constitute invariance under anti-unitary transformations (whose classical analogue are called anticanonical). For a particle in a magnetic field B plus scalar potential V, an example is TSx where Sx is a mirror reflection under which B and V are invariant. The authors illustrate this numerically for a single flux line in a hard-walled enclosure (Aharonov-Bohm quantum billiards), which also provides an example of an anti-unitary symmetry of non-geometrical origin; the spectral fluctuations are, as predicted, GOE rather than GUE.

Quantising a generic family of billiards with analytic boundaries

A generic family of plane billiards has been discovered recently. The shape of the boundary is given by the quadratic conformal image of the unit circle, and is thus real analytic. For small deformations of the unit disc the billiard is a typical KAM system, but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point. The Kolmogorov entropy has been calculated, and it increases with the deformation of the boundary. The author studies aspects of the quantum chaos for this billiard. He solves numerically the eigenvalue problem for the Laplace operator with Dirichlets boundary condition. He examines the spectrum, and inspects the avoided crossings at which mixing of nearby states occurs. The variation of the nodal structure and of the localisation properties of the eigenfunctions is studied. In analysing the level spacing distribution he finds a continuous transition from the Poisson distribution towards the Wigner distribution. The exponent in the level repulsion law varies continuously along with a generic perturbation. For small perturbations it seems to be proportional to the square root of the perturbation parameter.

The algebraic quantisation of the Birkhoff-Gustavson normal form

The author develops an algebraic quantisation method for the Birkhoff-Gustavson normal form. For this purpose the Weyl quantisation rule is used. The method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities. The author gives a brief review of the normal form, derives some of its general properties, and finds a general analytic solution for the fourth-degree normal form for Hamiltonians of two degrees of freedom. In particular, this includes the Henon-Heiles system. The author compares the results of specific examples with other works. The question of canonically invariant quantisation, the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed. The author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals. If the normal form accidentally terminates, so that the classical system is integrable, then this implies quantum integrability of the normal-form Hamiltonian.

Energy level statistics in the transition region between integrability and chaos

A generic one-parameter family of billiards discovered and introduced by Robnik (1983) is used to study the spectral properties of corresponding quantum systems. When the parameter is varied a smooth transition from an integrable system over a typical KAM system to an almost ergodic system can be observed. The authors calculate up to 7600 lowest reliable energy levels. A detailed analysis of the numerical data shows significant deviation from the semiclassical Berry-Robnik formulae for the nearest-neighbour level spacing distribution P(S) except for large level spacings, S>1, which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct. At small S the power-law level repulsion is clearly observed and a fit by the phenomenological formula by Izrailev (1988,1989) is statistically significant.

Energy transport and detailed verification of Fourier heat law in a chain of colliding harmonic oscillators

The authors study a simple nonlinear classical Hamiltonian system with positive K-entropy, a model for heat conduction, and they find that it obeys the Fourier heat law. Numerical simulation of its dynamics can be performed very efficiently, so they are able to explore it in detail. They verify the Fourier heat law and calculate the coefficient of thermal conductivity K by three independent methods. The first is direct simulation, i.e. simulating the dynamics of the chain between two heat reservoirs. The second is the Green-Kubo formalism which is derived in a self-contained manner. The third method-the one-sided heating of a semi-infinite cold chain-is new and gives the best results. It yields the entire temperature dependence K(T) in a single numerical simulation and definitely demonstrates the validity of the Fourier heat law at all temperatures for the given system. They believe that this method can also be useful for other systems. They derive analytically the asymptotic behaviour of the coefficient of thermal conductivity at low temperatures T to 0 and observe that it agrees with numerical results obtained by the Green-Kubo formalism, which gives by far the best results at very low temperatures.

Semiclassical energy level statistics in the transition region between integrability and chaos: transition from Brody-like to Berry-Robnik behaviour

We study the energy level statistics of the generic Hamiltonian systems in the transition region between integrability and chaos and present the theoretical and numerical evidence that in the ultimate (far) semiclassical limit the Berry-Robnik (1984) approach is the asymptotically exact theory. However, before reaching that limit, one observes phenomenologically a quasi-universal behaviour characterized by the fractional power-law level repulsion and globally quite well described by the Brody (or Izrailev) distribution. We offer theoretical arguments explaining this extremely slow transition and demonstrate it numerically in improved statistics of the Robnik billiard and in the standard (Chirikov) map on a torus.

Gravity trapping on a finite thickness domain wall: An analytic study

We construct an explicit model of the gravity trapping domain-wall potential, where for the first time we can study explicitly the graviton wave function fluctuations for any thickness of domain wall. A concrete form of the potential depends on one parameter 0{ 1, for which the fluctuation modes exhibit a resonance behavior, and which could sizably affect the modifications of the four-dimensional Newtons law at distances that typically are by 4 orders of magnitude larger than those relevant for Newtons law modifications of thin walls.

In-flight dissipation as a mechanism to suppress Fermi acceleration.

Some dynamical properties of time-dependent driven elliptical-shaped billiards are studied. It was shown that for conservative time-dependent dynamics the model exhibits Fermi acceleration [Phys. Rev. Lett. 100, 014103 (2008).] On the other hand, it was observed that damping coefficients upon collisions suppress such a phenomenon [Phys. Rev. Lett. 104, 224101 (2010)]. Here, we consider a dissipative model under the presence of in-flight dissipation due to a drag force which is assumed to be proportional to the square of the velocity of the particle. Our results reinforce that dissipation leads to a phase transition from unlimited to limited energy growth. The behavior of the average velocity is described using scaling arguments.