arXiv subjects

Marko Robnik

Publications and source records attributed to Marko Robnik.

At least 37 records · Page 2Linked to original sources

arXiv2021

Effects of stickiness in the classical and quantum ergodic lemon billiard

We study the classical and quantum ergodic lemon billiard introduced by Heller and Tomsovic in Phys. Today 46(7), 38 (1993), for the case $B=1/2$, which is a classically ergodic system (without a rigorous proof) exhibiting strong stickiness regions around a zero-measure bouncing ball modes. The structure of the classical stickiness regions is uncovered in the S-plots introduced by Lozej [Phys. Rev. E 101, 052204 (2020)]. A unique classical transport or diffusion time cannot be defined. As a consequence the quantum states are characterized by the following nonuniversal properties: (i) All eigenstates are chaotic but localized as exhibited in the Poincaré-Husimi (PH) functions. (ii) The entropy localization measure A (also the normalized inverse participation ratio) has a nonuniversal distribution, typically bimodal, thus deviating from the beta distribution, the latter one being characteristic of uniformly chaotic systems with no stickiness regions. (iii) The energy-level spacing distribution is Berry-Robnik-Brody (BRB), capturing two effects: the quantally divided phase space (because most of the PH functions are either the inner-ones or the outer-ones, dictated by the classical stickiness, with an effective parameter $μ_1$ measuring the size of the inner region bordered by the sticky invariant object, namely, a cantorus), and the localization of PH functions characterized by the level repulsion (Brody) parameter $β$. (iv) In the energy range considered (between 20 000 states to 400 000 states above the ground state) the picture (the structure of the eigenstates and the statistics of the energy spectra) is not changing qualitatively, as $β$ fluctuates around 0.8, while $μ_1$ decreases almost monotonically, with increasing energy.

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arXiv2021

Aspects of diffusion in the stadium billiard

We perform a detailed numerical study of diffusion in the $\varepsilon$ stadium of Bunimovich, and propose an empirical model of the local and global diffusion for various values of $\varepsilon$ with the following conclusions: (i) the diffusion is normal for all values of $\varepsilon \leq(0.3)$ and all initial conditions, (ii) the diffusion constant is a parabolic function of the momentum (i.e., we have inhomogeneous diffusion), (iii) the model describes the diffusion very well including the boundary effects, (iv) the approach to the asymptotic equilibrium steady state is exponential, (v) the so-called random model (Robnik et al., 1997) is confirmed to apply very well, (vi) the diffusion constant extracted from the distribution function in momentum space and the one derived from the second moment agree very well. The classical transport time, an important parameter in quantum chaos, is thus determined.

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arXiv2021

Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system

We perform a detailed study of the chaotic component in mixed-type Hamiltonian systems on the example of a family of billiards [introduced by Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)]. The phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times. The structure of the chaotic component is discerned from the cells visited by the chaotic orbit. The fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box-counting method. The cell-filling dynamics is compared to a model of uncorrelated motion, the so-called random model [Robnik et al. J. Phys. A: Math. Gen. 30, L803 (1997)], and deviations attributed to sticky objects in the phase space are found. The statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run. The stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times. The recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed. The standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale. Three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter. Lastly, the decay of correlations and the diffusion of momenta is analyzed.

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arXiv2015

Studies of dynamical localization in a finite-dimensional model of the quantum kicked rotator

We review our recent works on the dynamical localization in the quantum kicked rotator (QKR) and the related properties of the classical kicked rotator (the standard map, SM). We introduce the Izrailev $N$-dimensional model of the QKR and analyze the localization properties of the Floquet eigenstates [{\em Phys. Rev. E} {\bf 87}, 062905 (2013)], and the statistical properties of the quasienergy spectra. We survey normal and anomalous diffusion in the SM, and the related accelerator modes [{\em Phys. Rev. E} {\bf 89}, 022905 (2014)]. We analyze the statistical properties [{\em Phys. Rev. E} {\bf 91},042904 (2015)] of the localization measure, and show that the reciprocal localization length has an almost Gaussian distribution which has a finite variance even in the limit of the infinitely dimensional model of the QKR, $N\rightarrow \infty$. This sheds new light on the relation between the QKR and the Anderson localization phenomenon in the one-dimensional tight-binding model. It explains the so far mysterious strong fluctuations in the scaling properties of the QKR. The reason is that the finite bandwidth approximation of the underlying Hamilton dynamical system in the Shepelyansky picture [{\em Phys. Rev. Lett.} {\bf 56}, 677 (1986)] does not apply rigorously. These results call for a more refined theory of the localization length in the QKR and in similar Floquet systems, where we must predict not only the mean value of the inverse of the localization length but also its (Gaussian) distribution. We also numerically analyze the related behavior of finite time Lyapunov exponents in the SM and of the $2\times2$ transfer matrix formalism.

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arXiv2014

Statistical properties of the energy in time-dependent homogeneous power law potentials

We study 1D Hamilton systems with homogeneous power law potential and their statistical behaviour, assuming the microcanonical distribution of the initial conditions and describing its change under monotonically increasing time-dependent function $a(t)$ (prefactor of the potential). Using the nonlinear WKB-like method by Papamikos and Robnik {\em J. Phys. A: Math. Theor. {\bf 44} (2012) 315102} and following a previous work by Papamikos G and Robnik M {\em J. Phys. A: Math. Theor. {\bf 45} (2011) 015206} we specifically analyze the mean energy, the variance and the adiabatic invariant (action) of the systems for large time $t\rightarrow\infty$ and we show that the mean energy and variance increase as powers of $a(t)$, while the action oscillates and finally remains constant. By means of a number of detailed case studies we show that the theoretical prediction is excellent which demonstrates the usefulness of the method in such applications.

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arXiv2014

Statistical properties of 1D parametrically kicked Hamilton systems

We study the 1D Hamilton systems and their statistical behaviour, assuming the initial microcanonical distribution and describing its change under a parametric kick, which by definition means a discontinuous jump of a control parameter of the system. Following a previous work by Papamikos and Robnik (J. Phys. A: Math. Theor. 44 (2012) 315102) we specifically analyze the change of the adiabatic invariant (the action) of the system under a parametric kick: A conjecture has been put forward that the change of the action at the mean energy always increases, which means, for the given statistical ensemble, that the entropy in the mean increases. By means of a detailed analysis of a great number of case studies we show that the conjecture largely is satisfied, except if either the potential is not smooth enough, or if the energy is too close to a stationary point of the potential (separatrix in the phase space). Very fast changes in a time dependent system quite generally can be well described by such a picture and by the approximation of a parametric kick, if the change of the parameter is sufficiently fast and takes place on the time scale of less than one oscillation period.

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arXiv2014

Survey on the role of accelerator modes for anomalous diffusion: The case of the standard map

We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis on understanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and their immediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is still much affected ("dragged") by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic approach rests upon the following applications: the GALI method to detect the regular and chaotic regions and thus to describe in detail the structure of the phase space, the description of the momentum distribution in terms of the Lévy stable distributions, the numerical calculation of the diffusion exponent and of the corresponding diffusion constant. We use this approach to analyze in detail and systematically the standard map at all values of the kick parameter $K$, up to $K=70$. All complex features of the anomalous diffusion are well understood in terms of the role of the accelerator modes, mainly of period 1 at large $K\ge 2π$, but also of higher periods (2,3,4,...) at smaller values of $K\le 2π$.

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arXiv2013

Dynamical localization in chaotic systems: spectral statistics and localization measure in the kicked rotator as a paradigm for time-dependent and time-independent systems

We study quantum kicked rotator in the classically fully chaotic regime, in the domain of the semiclassical behaviour. We use Izrailev's N-dimensional model for various N<=4000, which in the limit N-> infinity tends to the quantized kicked rotator, not only for K=5 as studied previously, but for many different values of the classical kick parameter 5<=K<=35, and also of the quantum parameter k. We describe the dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent fully chaotic or/and mixed type Hamilton systems. We generalize the localization length L and the scaling variable (L/N) to the case of anomalous classical diffusion. We study the generalized classical diffusion also in the regime where the simple minded theory of the normal diffusion fails. We greatly improve the accuracy of the numerical calculations with the following conclusions: The level spacings distribution of the eigenphases is very well described by the Brody distribution, systematically better than by other proposals, for various Brody exponents. When N->infinity and L is fixed we have always Poisson, even in fully chaotic regime. We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure describing the degree of dynamical localization of the eigenfunction. The resulting localization parameter is found to be almost equal to the Brody parameter. We show the existence of a scaling law between the localization parameter and the scaling variable L/N, now including the regimes of anomalous diffusion. The above findings are important also in time-independent Hamilton systems, like in mixed type billiards, where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates.

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arXiv2013

Quantum localization of chaotic eigenstates and the level spacing distribution

The phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos (or wave chaos), and thus plays an important role in general quantum mechanics or even in general wave mechanics. In this work we propose two different localization measures characterizing the degree of quantum localization, and study their relation to another fundamental aspect of quantum chaos, namely the (energy) spectral statistics. Our approach and method is quite general, and we apply it to billiard systems. One of the signatures of the localization of chaotic eigenstates is a fractional power-law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance $S$ goes like $\propto S^β$ for small $S$, where $0 \leq β\leq 1$, and $β= 1$ corresponds to completely extended states. We show that there is a clear functional relation between the exponent β and the two different localization measures. One is based on the information entropy and the other one on the correlation properties of the Husimi functions. We show that the two definitions are surprisingly linearly equivalent. The approach is applied in the case of a mixed-type billiard system (Robnik 1983), in which the separation of regular and chaotic eigenstates is performed.

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arXiv2013

Dynamical localization of chaotic eigenstates in the mixed-type systems: spectral statistics in a billiard system after separation of regular and chaotic eigenstates

We study the quantum mechanics of a billiard (Robnik 1983) in the regime of mixed-type classical phase space (the shape parameter λ=0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincaré Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all to be dynamically localized, meaning that they do not occupy unformly the entire available chaotic classical phase space component, but are localized on a proper subset. We find with unprecedented precision and statistical significance that the level spacing distribution of the regular levels obeys the Poisson statistics, and the chaotic ones obey the Brody statistics, as anticipated in a recent paper by Batistić and Robnik (2010), where the entire spectrum was found to obey the BRB statistics. There are no effects of dynamical tunneling in this regime, due to the high energies, as they decay exponentially with the inverse effective Planck constant which is proportional to the square root of the energy.

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arXiv2013

The intermediate level statistics in dynamically localized chaotic eigenstates

We demonstrate that the energy or quasienergy level spacing distribution in dynamically localized chaotic eigenstates is excellently described by the Brody distribution, displaying the fractional power law level repulsion. This we show in two paradigmatic systems, namely for the fully chaotic eigenstates of the kicked rotator at K=7, and for the chaotic eigenstates in the mixed-type billiard system (Robnik 1983), after separating the regular and chaotic eigenstates by means of the Poincaré Husimi function, at very high energies with great statistical significance (587654 eigenstates, starting at about 1.000.000 above the ground state). This separation confirms the Berry-Robnik picture, and is performed for the first time at such high energies.

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arXiv2013

Dynamical localization in kicked rotator as a paradigm of other systems: spectral statistics and the localization measure

We study the intermediate statistics of the spectrum of quasi-energies and of the eigenfunctions in the kicked rotator, in the case when the corresponding system is fully chaotic while quantally localized. As for the eigenphases, we find clear evidence that the spectral statistics is well described by the Brody distribution, notably better than by the Izrailev's one, which has been proposed and used broadly to describe such cases. We also studied the eigenfunctions of the Floquet operator and their localization. We show the existence of a scaling law between the repulsion parameter with relative localization length, but only as a first order approximation, since another parameter plays a role. We believe and have evidence that a similar analysis applies in time-independent Hamilton systems.

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arXiv2011

Dynamical properties of a particle in a wave packet: scaling invariance and boundary crisis

Some dynamical properties present in a problem concerning the acceleration of particles in a wave packet are studied. The dynamics of the model is described in terms of a two-dimensional area preserving map. We show that the phase space is mixed in the sense that there are regular and chaotic regions coexisting. We use a connection with the standard map in order to find the position of the first invariant spanning curve which borders the chaotic sea. We find that the position of the first invariant spanning curve increases as a power of the control parameter with the exponent 2/3. The standard deviation of the kinetic energy of an ensemble of initial conditions obeys a power law as a function of time, and saturates after some crossover. Scaling formalism is used in order to characterize the chaotic region close to the transition from integrability to nonintegrability and a relationship between the power law exponents is derived. The formalism can be applied in many different systems with mixed phase space. Then, dissipation is introduced into the model and therefore the property of area preservation is broken, and consequently attractors are observed. We show that after a small change of the dissipation, the chaotic attractor as well as its basin of attraction are destroyed, thus leading the system to experience a boundary crisis. The transient after the crisis follows a power law with exponent -2.

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arXiv2011

Statistical properties of a dissipative kicked system: critical exponents and scaling invariance

A new universal {\it empirical} function that depends on a single critical exponent (acceleration exponent) is proposed to describe the scaling behavior in a dissipative kicked rotator. The scaling formalism is used to describe two regimes of dissipation: (i) strong dissipation and (ii) weak dissipation. For case (i) the model exhibits a route to chaos known as period doubling and the Feigenbaum constant along the bifurcations is obtained. When weak dissipation is considered the average action as well as its standard deviation are described using scaling arguments with critical exponents. The universal {\it empirical} function describes remarkably well a phase transition from limited to unlimited growth of the average action.

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arXiv2011

Scaling Invariance in a Time-Dependent Elliptical Billiard

We study some dynamical properties of a classical time-dependent elliptical billiard. We consider periodically moving boundary and collisions between the particle and the boundary are assumed to be elastic. Our results confirm that although the static elliptical billiard is an integrable system, after to introduce time-dependent perturbation on the boundary the unlimited energy growth is observed. The behaviour of the average velocity is described using scaling arguments.

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arXiv2010

In-flight dissipation as a mechanism to suppress Fermi acceleration

Some dynamical properties of time-dependent driven elliptical-shaped billiard are studied. It was shown that for the conservative time-dependent dynamics the model exhibits the Fermi acceleration [Phys. Rev. Lett. 100, 014103 (2008)]. On the other hand, it was observed that damping coefficients upon collisions suppress such 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 particle's velocity. Our results reinforce that dissipation leads to a phase transition from unlimited to limited energy growth. The behaviour of the average velocity is described using scaling arguments.

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arXiv2008

Gravity Trapping on a Finite Thickness Domain Wall: An Analytic Study

We construct an explicit model of gravity trapping domain wall potential where for the first time we can study explicitly the graviton wave function fluctuations for any thickness of the domain wall. A concrete form of the potential depends on one parameter 0 <= x <= pi/2, which effectively parameterizes the thickness of the domain wall with specific limits x -> 0 and x -> pi/2 corresponding to the thin and the thick wall, respectively. The analysis of the continuum Kaluza Klein fluctuations yields explicit expressions for both the small and large Kaluza Klein energy. We also derive specific explicit conditions in the regime x > 1, for which the fluctuation modes exhibit a resonance behaviour, and which could sizably affect the modifications of the four-dimensional Newton's law at distances, which typically are by four orders of magnitude larger than those relevant for Newton's law modifications of thin walls.

hep-th↗

arXiv2006

Expanded boundary integral method and chaotic time-reversal doublets in quantum billiards

We present the expanded boundary integral method for solving the planar Helmholtz problem, which combines the ideas of the boundary integral method and the scaling method and is applicable to arbitrary shapes. We apply the method to a chaotic billiard with unidirectional transport, where we demonstrate existence of doublets of chaotic eigenstates, which are quasi-degenerate due to time-reversal symmetry, and a very particular level spacing distribution that attains a chaotic Shnirelman peak at short energy ranges and exhibits GUE-like statistics for large energy ranges. We show that, as a consequence of such particular level statistics or algebraic tunneling between disjoint chaotic components connected by time-reversal operation, the system exhibits quantum current reversals.

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