1. Quantum-classical correspondence between quantum chaos and finite-time classical dynamicsQian Wang, Marko Robnik, 2025, original scientific article Abstract: Although the importance of quantum-classical correspondence has been recognized in numerous studies of quantum chaos, its usefulness in understanding quantum chaos through finite-time classical dynamics remains less well understood.We address this question in this work by performing a detailed analysis of how the quantum chaotic measure relates to the chaoticity of the finite-time classical trajectories. A good correspondence between them has been revealed in time- dependent and many-body systems, both of them being of the mixed type. In particular, we show that the dependence of the quantum chaotic measure on the chaoticity of finite-time trajectories can be well captured by a function that is independent of the system. This strongly implies the universal validity of the finite-time quantum-classical correspondence. Our findings enhance the understanding of quantum-classical correspondence and provide a promising approach to explore the ergodic hierarchy in quantum systems.
Keywords: quantum chaos, finite-time classical dynamics, quantum-classical correspondence, dynamical chaos, mixed type, kicked top model, Dicke model
Published in DKUM: 03.11.2025; Views: 0; Downloads: 0
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2. Multifractality in quasienergy space of coherent states as a signature of quantum chaosQian Wang, Marko Robnik, 2021, original scientific article Abstract: We present the multifractal analysis of coherent states in kicked top model by expanding them in the basis of Floquet operator eigenstates. We demonstrate the manifestation of phase space structures in the multifractal properties of coherent states. In the classical limit, the classical dynamical map can be constructed, allowing us to explore the corresponding phase space portraits and to calculate the Lyapunov exponent. By tuning the kicking strength, the system undergoes a transition from regularity to chaos. We show that the variation of multifractal dimensions of coherent states with kicking strength is able to capture the structural changes of the phase space. The onset of chaos is clearly identified by the phase-space-averaged multifractal dimensions, which are well described by random matrix theory in a strongly chaotic regime. We further investigate the probability distribution of expansion coefficients, and show that the deviation between the numerical results and the prediction of random matrix theory behaves as a reliable detector of quantum chaos.
Keywords: quantum chaos, multifractal analysis, kicked top, coherent states
Published in DKUM: 13.10.2023; Views: 433; Downloads: 35
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3. Fluctuating number of energy levels in mixed-type lemon billiardsČrt Lozej, Dragan Lukman, Marko Robnik, 2021, original scientific article Abstract: In this paper, the fluctuation properties of the number of energy levels (mode fluctuation) are studied in the mixed-type lemon billiards at high lying energies. The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between the centers, as introduced by Heller and Tomsovic. In this paper, the case of two billiards, defined by B = 0.1953, 0.083, is studied. It is shown that the fluctuation of the number of energy levels follows the Gaussian distribution quite accurately, even though the relative fraction of the chaotic part of the phase space is only 0.28 and 0.16, respectively. The theoretical description of spectral fluctuations in the Berry-Robnik picture is discussed. Also, the (golden mean) integrable rectangular billiard is studied and an almost Gaussian distribution is obtained, in contrast to theory expectations. However, the variance as a function of energy, E, behaves as - E, in agreement with the theoretical prediction by Steiner.
Keywords: nonlinear dynamics, quantum chaos, mixed-type systems, energy level statistics, lemon billiards, billiards
Published in DKUM: 13.10.2023; Views: 518; Downloads: 27
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4. Phenomenology of quantum eigenstates in mixed-type systems: Lemon billiards with complex phase space structureČrt Lozej, Dragan Lukman, Marko Robnik, 2022, original scientific article Abstract: The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic [E. J. Heller and S. Tomsovic, Phys. Today 46, 38 (1993)]. This paper is a continuation of our recent papers on a classical and quantum ergodic lemon billiard (B = 0.5) with strong stickiness effects [C. Lozej ˇ et al., Phys. Rev. E 103, 012204 (2021)], as well as on the three billiards with a simple mixed-type phase space and no stickiness [C. Lozej ˇ et al., Nonlin. Phenom. Complex Syst. 24, 1 (2021)]. Here we study two classical and quantum lemon billiards, for the cases B = 0.1953, 0.083, which are mixed-type billiards with a complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently [ C. Lozej, D. Lukman, and M. ˇ Robnik, Physics 3, 888 (2021)]. We calculate a very large number (106) of consecutive eigenstates and their Poincaré-Husimi (PH) functions, and analyze their localization properties by studying the entropy localization measure and the normalized inverse participation ratio. We introduce an overlap index, which measures the degree of the overlap of PH functions with classically regular and chaotic regions. We observe the existence of regular states associated with invariant tori and chaotic states associated with the classically chaotic regions, and also the mixed-type states. We show that in accordance with the Berry-Robnik picture and the principle of uniform semiclassical condensation of PH functions, the relative fraction of mixed-type states decreases as a power law with increasing energy, thus, in the strict semiclassical limit, leaving only purely regular and chaotic states. Our approach offers a general phenomenological overview of the structural and localization properties of PH functions in quantum mixed-type Hamiltonian systems.
Keywords: quantum physics, energy, localization, quantum chaos, billiards, chaotic systems
Published in DKUM: 12.10.2023; Views: 288; Downloads: 26
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6. Transport and Localization in Classical and Quantum BilliardsČrt Lozej, 2020, doctoral dissertation Abstract: In this thesis the classical and quantum dynamics in billiard systems are considered. Extensive numerical studies of the classical transport properties in several examples of billiard families including the ergodic Bunimovich stadium and cut-circle billiards and the mixed-type Robnik and lemon billiards are performed. The analysis of the transport is based on the random model of diffusion which assumes that due the strongly chaotic dynamics the motion of the orbit on the discretized phase space is temporally uncorrelated. The cause of the deviations from the random model dynamics is traced to dynamical trapping due to stickiness. A novel approach to locally quantifying stickiness based on the statistics of the recurrence times is presented and applied to distinguish between exponential decays of recurrence times and other types of decays. This enables the identification of sticky areas in the chaotic components. Detailed maps of their structure for a wide range of parameter values, mapping the evolution of the mixed-phase spaces and revealing some particularly interesting special examples are presented. The recurrence time distributions in sticky areas are found to be well described by a mixture of exponential decays. The transport of particle ensembles in the momentum space of classical billiards is described by using an inhomogeneous diffusion model and the classical transport times are determined. The classical transport times are vital for the analysis of the localization of chaotic eigenstates in quantum billiards. The control parameter that describes the the degree of localization of the chaotic quantum eigenstates is the ratio between the Heisenberg time (Planck's constant divided by the mean level spacing) and the classical transport time. Extensive numerical calculations of the high-lying spectra and eigenstates of the stadium, Robnik and lemon quantum billiards are performed. The spectral statistics are analysed in terms of the standard methods of quantum chaos. The level repulsion exponent of localized eigenstates is found to be a rational function of the control parameter. The degree of localization is determined with respect to localization measures based on the Poincaré-Husimi representation of the eigenstates. The mean localization measure is found to be a rational function of the control parameter and linearly related to the level repulsion exponent. The distributions of the localization measures are analysed and found to be of a universal shape well described by a two parameter empirical distribution in billiards with no apparent stickiness. The nonuniversal system specific features of localization measure distributions are related to the presence of sticky areas in the phase spaces of classical billiards with specific examples shown.
Keywords: Transport, localization, chaos, quantum chaos, Hamiltonian systems, level spacing distribution, mixed phase space, billiard, quantum billiard, Husimi functions, stickiness, cantorus, chaotic eigenstates, level repulsion.
Published in DKUM: 13.01.2021; Views: 1559; Downloads: 169
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7. Organizational learning supported by machine learning models coupled with general explanation methods : a case of B2B sales forecastingMarko Bohanec, Marko Robnik Šikonja, Mirjana Kljajić Borštnar, 2017, original scientific article Abstract: Background and Purpose: The process of business to business (B2B) sales forecasting is a complex decision-making process. There are many approaches to support this process, but mainly it is still based on the subjective judgment of a decision-maker. The problem of B2B sales forecasting can be modeled as a classification problem. However, top performing machine learning (ML) models are black boxes and do not support transparent reasoning. The purpose of this research is to develop an organizational model using ML model coupled with general explanation methods. The goal is to support the decision-maker in the process of B2B sales forecasting.
Design/Methodology/Approach: Participatory approach of action design research was used to promote acceptance of the model among users. ML model was built following CRISP-DM methodology and utilizes R software environment.
Results: ML model was developed in several design cycles involving users. It was evaluated in the company for several months. Results suggest that based on the explanations of the ML model predictions the users’ forecasts improved. Furthermore, when the users embrace the proposed ML model and its explanations, they change their initial beliefs, make more accurate B2B sales predictions and detect other features of the process, not included in the ML model.
Conclusions: The proposed model promotes understanding, foster debate and validation of existing beliefs, and thus contributes to single and double-loop learning. Active participation of the users in the process of development, validation, and implementation has shown to be beneficial in creating trust and promotes acceptance in practice.
Keywords: decision support, organizational learning, machine learning, explanations
Published in DKUM: 01.09.2017; Views: 1751; Downloads: 373
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8. Expanded boundary integral method and chaotic time-reversal in quatum billiardsGregor Veble, Tomaž Prosen, Marko Robnik, 2007, original scientific article Abstract: 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 the 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 Gaussian Unitary Ensemble (GUE) like statistics for large energy ranges. We show that, as a consequence of such particular level statistics or algebraic tunnelling between disjoint chaotic components connected by time-reversal operation, the system exhibits quantum current reversals.
Keywords: 2D Helmholtz equation, energy spectrum, quantum billiard
Published in DKUM: 03.07.2017; Views: 1811; Downloads: 221
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9. Model napovedovanja prodajnih priložnosti v medorganizacijskem poslovanju z uporabo metod strojnega učenjaMarko Bohanec, 2017, doctoral dissertation Abstract: Področje medorganizacijske (B2B) prodaje je zahtevno. Običajno ne predvideva le prodaje končnih izdelkov, temveč so predmet prodaje kompleksnejše rešitve, prilagojene kupcu. Kupci pri tem sledijo svojim internim procesom, želijo doseči prilagoditve elementov pogodbe, se pogajajo in podobno. To zahteva od prodajalcev dobro poznavanje pričakovanj strank, njihovih želja in potreb.
Proces B2B prodaje je zato daljši in kompleksnejši. V disertaciji se osredotočamo na napovedovanje prodajnega izida v medorganizacijski prodaji, ki v praksi večinoma temelji na subjektivni presoji prodajalcev. Glede na napovedi prodajalcev se podjetja odločajo o virih in aktivnostih, zato netočne napovedi lahko vodijo v nepopravljive posledice. Raziskave so pokazale, da podjetja, ki svoje odločitve temeljijo na podatkih (angl. ``data driven decison-making''), izkazujejo boljše poslovne rezultate. Vendar pa raziskave kažejo tudi, da je uporaba metod in orodij, ki temeljijo na podatkih, v praksi še vedno šibka. To lahko pripišemo slabemu razumevanju metod in orodij za podporo odločanju ter tudi nezaupanju v tehnologijo.
Motivacija za raziskavo izhaja iz zaznanega problemskega stanja v medorganizacijski prodaji in vrzeli, ki smo jo zaznali v akademski literaturi. Izhajamo iz teze, da je mogoče z uporabo modelov stojnega učenja pomagati prodajalcu in podjetju tako, da pri napovedovanju poslovnega izida delajo manj napak. V ta namen smo po metodologiji akcijske raziskave in razvoja (angl. ``action design research'', s kratico ADR) razvili model napovedovanja prodajnih priložnosti v medorganizacijskem poslovanju z uporabo metod strojnega učenja. Pri tem sta nastala dva artefakta: informacijsko-tehnološki (IT) artefakt, ki temelji na modelih strojnega učenja, podkrepljenih s transparentnimi razlagami, ter organizacijski artefakt, ki spodbuja vključevanje spoznanj iz IT-artefakta v proces napovedovanja in organizacijsko učenje. Prednost ADR je v tem, da vključuje uporabnike v razvoj in evalvacijo modela že na začetku raziskave. Na ta način uporabniki lahko izrazijo svoja pričakovanja, sproti vrednotijo model ter tudi sproti predlagajo spremembe. To krepi zaupanje v razvit model in povečuje zavezanost h kasnejši uporabi v praksi.
Jedro disertacije predstavlja model napovedovanja prodajnih priložnosti v medorganizacijskem poslovanju z uporabo metod strojnega učenja. Metode strojnega učenja se iz prodajne zgodovine naučijo prepoznati značilnosti prodaje. Ko se pojavijo nove priložnosti, metode ocenijo njihovo zrelost in ponudijo odločevalcem razlago vplivnih dejavnikov, omogočajo pa tudi analizo vpliva različnih prodajnih aktivnosti s pomočjo ``kaj-če'' analize.
Pri tem uporabljamo poenoten format napovedi in njihovih razlag, ki podpirajo različne modele. Tako omogočamo uporabo visoko zmogljivih metod strojnega učenja (npr. naključni gozd), ki so običajno zaradi svoje zapletenosti netransparentne in jim uporabniki stežka zaupajo.
Da smo lahko razvili model, smo opravili dodatne raziskave za oblikovanje nabora atributov, ki opisujejo proces B2B prodaje in razvili optimizacijske postopke za detekcijo šuma in redundance atributov. Za učinkovito detekcijo kvalitete učne množice smo razvili vizualno metodo.
Potrdili smo domnevo, da je želen organizacijsko-informacijski model mogoče zgraditi, saj je večina uporabljenih metod dosegla klasifikacijsko točnost nad 70\%. Za podrobnejšo analizo vpliva atributov smo razvili simulacijske in optimizacijske algoritme. Praksa potrjuje koristnost razvitega modela, saj se je v realnem podjetju z uporabo modela točnost napovedi bistveno izboljšala. S kombinacijo uporabe modelov, znanja in prakse ekspertov, smo tako prispevali k preseganju pomanjkljivosti posameznih pristopov.
Uporabljene metode predstavljajo novost na področju organizacijskih znanosti in tako prispevajo k znanstveni literaturi na področju organizacijskega učenja in uporabe metod strojnega učenja.
Keywords: strojno učenje, medorganizacijska prodaja, organizacijsko učenje, napovedovanje izida, razlaga modelov, analiza kaj-če
Published in DKUM: 24.04.2017; Views: 1904; Downloads: 287
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10. Dynamical and statistical properties of time-dependent one-dimensional nonlinear Hamilton systemsDimitrios Andresas, 2015, doctoral dissertation Abstract: We study the one-dimensional time-dependent Hamiltonian systems and their statistical behaviour, assuming the microcanonical ensemble of initial conditions and describing the evolution of the energy distribution in three characteristic cases: 1) parametric kick, which by definition means a discontinuous jump of a control parameter of the system, 2) linear driving, and 3) periodic driving. For the first case we specifically analyze the change of the adiabatic invariant (the canonical action) of the system under a parametric kick: A conjecture has been put forward by Papamikos and Robnik (2011) that the action at the mean energy always increases, which means, for the given statistical ensemble, that the Gibbs entropy in the mean increases (PR property). By means of a detailed rigorous 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 (e.g. has discontinuous first derivative), or if the energy is too close to a stationary point of the potential (separatrix in the phase space). We formulate the conjecture in full generality, and perform the local theoretical analysis by introducing the ABR property. For the linear driving we study first 1D Hamilton systems with homogeneous power law potential and their statistical behaviour under monotonically increasing time-dependent function A(t) (prefactor of the potential). We used the nonlinear WKB-like method by Papamikos and Robnik J. Phys. A: Math. Theor., 44:315102, (2012) and following a previous work by Papamikos G and Robnik M J. Phys. A: Math. Theor., 45:015206, (2011) we specifically analyze the mean energy, the variance and the
adiabatic invariant (action) of the system for large time t→∞. We also show analytically that the mean energy and the 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 correct. For the periodic driving cases we study the 1D periodic quartic oscillator and its statistical behaviour under periodic time-dependent function A(t) (prefactor of the potential). We compare the results for three different drivings, the periodic parametrically kicked case (discontinuous jumps of $A(t)$), the piecewise linear case (sawtooth), and the smooth case (harmonic). Considering the Floquet map and the energy distribution we perform careful numerical analysis using the 8th order symplectic integrator and present the phase portraits for each case, the evolution of the average energy and the distribution function of the final energies. In the case where we see a large region of chaos connected to infinity, we indeed find escape orbits going to infinity, meaning that the energy growth can be unbounded, and is typically exponential in time.
The main results are published in two papers:
Andresas, Batistić and Robnik Phys. Rev. E, 89:062927, (2014) and
Andresas and Robnik J. Phys. A: Math. Theor., 47:355102, (2014).
Keywords: one-dimensional nonlinear Hamiltonian systems, adiabatic invariant, parametric kick, periodic driving, linear driving, energy distribution, WKB method, action
Published in DKUM: 02.03.2015; Views: 3309; Downloads: 132
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