Martin Horvat

University of Ljubljana · Department of Physics

The reduced-complexity models developed by Edward Lorenz are widely used in atmospheric and climate sciences to study nonlinear aspect of dynamics and to demonstrate new methods for numerical weather prediction. A set of inviscid Lorenz models describing the dynamics of a single variable in a zonally-periodic domain, without dissipation and forcing...

The performance of estimated models is often evaluated in terms of their predictive capability. In this study, we investigate another important aspect of estimated model evaluation: the disparity between the statistical and dynamical properties of estimated models and their source system. Specifically, we focus on estimated models obtained via the...

The performance of estimated models is often evaluated in terms of their predictive capability. In this study, we investigate another important aspect of estimated model evaluation: the disparity between the statistical and dynamical properties of estimated models and their source system. Specifically, we focus on estimated models obtained via the...

The goal of this paper is to present the results of an ESA project, which aimed at experimentally test the idea of an autonomous relativistic positioning system (RPS). In a number of experiments, we used pseudo-range data from the Sentinel3A satellite. The data was used in an RPS simulator to retrieve the relative orbital elements of the satellite....

The time-dependent fields obtained by solving partial differential equations in two and more dimensions quickly overwhelm the analytical capabilities of the human brain. A meaningful insight into the temporal behaviour can be obtained by using scalar reductions, which, however, come with a loss of spatial detail. Dynamic Mode Decomposition is a dat...

In complex dynamical systems, the detection of coupling and its direction from observed time series is a challenging task. We study coupling in coupled Duffing oscillator systems in regular and chaotic dynamical regimes. By observing the conditional mutual information (CMI) based on the Shannon entropy, we successfully infer the direction of coupli...

We present a theoretical method for calculating multiphoton ionization amplitudes and cross sections of few-electron atoms. The present approach is based on an extraction of partial wave amplitudes from a scattering wave function, which is calculated by solving a system of driven Schrödinger equations. The extraction relies on a description of part...

We present a theoretical method for calculating multiphoton ionization amplitudes and cross sections of few-electron atoms. The present approach is based on an extraction of partial wave amplitudes from a scattering wave function, which is calculated by solving a system of driven Schroedinger equations. The extraction relies on a description of par...

In complex dynamical systems, the detection of coupling and its direction from observed time series is a challenging task. We study coupling in coupled Duffing oscillator systems in regular and chaotic dynamical regimes. By observing the conditional mutual information (CMI) based on the Shannon entropy, we successfully infer the direction of coupli...

Traditionally, the effects of interstellar extinction on binary star light curves have been treated as a uniform reduction in the observed brightness of the system that is independent of the orbital phase. However, unless the orbital plane of the system coincides with the plane of the sky, or if the two stars are completely identical and present wi...

The paper presents a method for the scale-dependent validation of the spatio-temporal variability in global weather or climate models and for their bias quantification in relation to dynamics. The method provides a relationship between the bias and simulated spatial and temporal variance by a model in comparison with verifying reanalysis data. For...

Traditionally, the effects of interstellar extinction on binary star light curves have been treated as a uniform reduction in the observed brightness of the system that is independent of orbital phase. However, unless the orbital plane of the system coincides with the plane of the sky, or if the two stars are completely identical and present with m...

A general framework for dealing with irradiation effects in the bolometric sense --- specifically, reflection with heat absorption and the consequent redistribution of the absorbed heat --- for systems of astrophysical bodies where the boundaries are used as support for the description of the processes, is presented. Discussed are its mathematical...

A general framework for dealing with irradiation effects in the bolometric sense is presented. Specifically, reflection with heat absorption and consequent redistribution of the absorbed heat, for systems of astrophysical bodies where the boundaries are used as support for the description of the processes. Discussed are its mathematical and physica...

Binary systems where the axis of rotation (spin) of one or both components is tilted w.r.t. the axis of revolution are called misaligned binary systems. The angle of misalignment, obliquity, has been measured for a handful of stars and extrasolar planets to date. Here, we present a mathematical framework for a complete and rigorous treatment of mis...

We record the outcomes of physical measurements as signals (sequences of values), where we are not interested in each value in particular but the characteristics of the signal as a whole. Signals can be analyzed in the statistical sense, where the time ordering of the data is irrelevant, or in the functional sense, where it becomes essential: then...

In this chapter we discuss methods for the solution of problems with ordinary differential equations, where we require the solution to satisfy the equation within the definition domain, and boundary conditions at its edges.

A time series or signal \(s(t)\in \Sigma \) should be understood as a sequential measurement of some quantity. The time variable and the corresponding signal may be discrete or continuous. The signals may be real, complex, or integer. In the analysis of time series we use mathematical tools to extract their basic characteristics and learn about the...

Solving partial differential equations (PDE) is so crucial to mathematical physics that we devote three chapters to it. The solution methods largely depend on the type of the PDE. The type of a general system of linear first-order equations $$ A{{\varvec{v}}}_x + B{{\varvec{v}}}_y = {{\varvec{c}}} \>, $$where \({{\varvec{v}}}=(v_1, v_2, \ldots , v_...

In finite difference methods the exact solution v of the differential equation is approximated by low-order polynomials interpolating v at several nearby mesh points. For example, ( 9.3) is an approximation of the derivative at \(x_j\) obtained by parabolic interpolation between \(x_{j-1}\), \(x_j\), and \(x_{j+1}\). The solution on the whole inter...

For physicist’s needs, numerical linear algebra is so comprehensively covered by classic textbooks (Blackford et al. ACM Trans Math Softw 28:135, 2002, [1]), (LAPACK, The LinearAlgebra PACKage, [2]) that another detailed description of the algorithms here would be pointless. To a larger extent than in other chapters we wish to merely set up road-si...

An ever increasing amount of computational work is being relegated to computers, and often we almost blindly assume that the obtained results are correct. At the same time, we wish to accelerate individual computation steps and improve their accuracy. Numerical computations should therefore be approached with a good measure of skepticism.

The basic concepts of difference methods for PDE in several dimensions are readily adopted from the discussion of one-dimensional initial-boundary-value problems (Chap. 9). We are seeking consistent, stable difference schemes and corresponding discretizations of the initial and boundary conditions by which we obtain convergence of the numerical sol...

In all areas of physics, mathematics, and engineering, we need to solve non-linear equations.

When we evaluate the expression \({{\varvec{f}}} = A{{\varvec{u}}}\), where \({{\varvec{u}}}\) and \({{\varvec{f}}}\) are vectors and A is a matrix, we solve a direct or forward problem. Given A we can precisely calculate \({{\varvec{f}}}\) for any \({{\varvec{u}}}\). The only danger we may anticipate in a computer is the one of over- or underflow,...

The Fourier transformation \(\mathcal{F}\) of the function f on the real axis is defined as \(F(\omega ) = \mathcal{F}[f](\omega ) = \int _{-\infty }^\infty f(x)\, \mathrm {e}^{-\mathrm {i}\omega x } \, \mathrm {d}x \>. \) The sufficient conditions for the existence of F are that f is absolutely integrable, i.e. \(\int _{-\infty }^\infty |f(x)| \,...

Often we try to understand the evolution of a system from a known initial to an unknown later state by observing the temporal variation (dynamics) of quantities that we use to describe the system. There are endless examples. Physicists study oscillations of non-linear mechanical pendula and electric circuits, monitor the motion of charged particles...

Binary systems where the axis of rotation (spin) of one or both components is tilted w.r.t. the axis of revolution are called misaligned binary systems. The angle of misalignment, obliquity, has been measured for a handful of stars and extrasolar planets to date. Here we present a mathematical framework for a complete and rigorous treatment of misa...

Contact binary stars have been known to have a peculiar and somewhat mysterious hydro- and thermodynamical structure since their discovery, which directly affects the radiation distribution in their atmospheres. Over the past several decades, however, observational data of contact binaries have been modeled through a simplified approach, involving...

It has long been recognized that the finite speed of light can affect the observed time of an event. For example, as a source moves radially toward or away from an observer, the path length and therefore the light travel time to the observer decreases or increases, causing the event to appear earlier or later than otherwise expected, respectively....

This book is intended to help advanced undergraduate, graduate, and postdoctoral students in their daily work by offering them a compendium of numerical methods. The choice of methods pays significant attention to error estimates, stability and convergence issues, as well as optimization of program execution speeds. Numerous examples are given throug...

We present a general framework for dealing with irradiation effects, i.e. reflection with heat absorption and consequent redistribution, integrated in PHysics Of Eclipsing BinariEs (PHOEBE) II and a proof-of-concept. A more detailed discussion on this topic will be presented in the forthcoming paper.

We assess the scale-dependent growth of forecast errors based on a 50-member global forecast ensemble from the European Centre for Medium Range Weather Forecasts. Simulated forecast errors are decomposed into scales and a new parametric model for the representation of the error growth is applied independently to every zonal wavenumber. In contrast...

The precision of photometric and spectroscopic observations has been systematically improved in the last decade, mostly thanks to space-borne photometric missions and ground-based spectrographs dedicated to finding exoplanets. The field of eclipsing binary stars strongly benefited from this development. Eclipsing binaries serve as critical tools fo...

The precision of photometric and spectroscopic observations has been systematically improved in the last decade, mostly thanks to space-borne photometric missions and ground-based spectrographs dedicated to finding exoplanets. The field of eclipsing binary stars strongly benefited from this development. Eclipsing binaries serve as critical tools fo...

Appendix S1. Derivation of heart rate variability and respiratory frequency variability. Appendix S2. Wavelet transform, wavelet phase coherence and synchronisation. Appendix S3. Automatic classification. Table S1. The parameters used in the vector‐based discriminatory analysis for the different subsets of data. Table S2. Five confusion matrice...

We present a variant of a Global Navigation Satellite System called a Relativistic Positioning System (RPS), which is based on emission coordinates. We modelled the RPS dynamics in a space-time around Earth, described by a perturbed Schwarzschild metric, where we included the perturbations due to Earth multipoles (up to the 6th), the Moon, the Sun,...

Depth of anaesthesia monitors usually analyse cerebral function with or without other physiological signals; non-invasive monitoring of the measured cardiorespiratory signals alone would offer a simple, practical alternative. We aimed to investigate whether such signals, analysed with novel, non-linear dynamic methods, would distinguish between the...

Current GNSS systems rely on global reference frames which are fixed to the Earth (via the ground stations) so their precision and stability in time are limited by our knowledge of the Earth dynamics. These drawbacks could be avoided by giving to the constellation of satellites the possibility of constituting by itself a primary and autonomous posi...

Current GNSS systems rely on global reference frames which are fixed to the Earth (via the ground stations) so their precision and stability in time are limited by our knowledge of the Earth dynamics. These drawbacks could be avoided by giving to the constellation of satellites the possibility of constituting by itself a primary and autonomous posi...

Current GNSS systems rely on global reference frames which are fixed to the Earth (via the ground stations) so their precision and stability in time are limited by our knowledge of the Earth dynamics. These drawbacks could be avoided by giving to the constellation of satellites the possibility of constituting by itself a primary and autonomous posi...

We propose a simple model of coupled heat and particle transport based on zero-dimensional classical deterministic dynamics, which is reminiscent of a railway switch whose action is a function only of the particle's energy. It is shown that already in the minimal three-terminal model, where the second terminal is considered as a probe with zero net...

We study transport properties of a disordered tight-binding model (XX spin chain) in the presence of dephasing. Focusing on diffusive behavior in the thermodynamic limit at high energies, we analytically derive the dependence of conductivity on dephasing and disorder strengths. As a function of dephasing, conductivity exhibits a single maximum at t...

Finite-difference methods for one-dimensional partial differential equations are introduced by first identifying the classes of equations upon which suitable discretizations are constructed. It is shown how parabolic equations and the corresponding boundary conditions are discretized such that a desired local order of error is achieved and that the...

This chapter first deals with various finite-difference methods for scalar boundary-value problems involving ordinary differential equations and for systems of such problems. The concepts of consistency, stability and convergence are introduced, as well as methods to increase the local solution precision by extrapolation. Shooting methods are offer...

The discrete Fourier transformation is one of the most important tools in the analysis of functions and signals, but its detailed numerical aspects are often disregarded or neglected. Fourier and sampling theorems, Parseval’s equality, and power spectral densities are introduced, and the concepts of signal uncertainty, aliasing, and leakage are dis...

Outcomes of physical measurements are most frequently recorded as signals that need to be processed statistically in order to infer their overall properties and features, and later compared to theoretical or phenomenological models. This chapter starts with an introduction to the basic statistical techniques of computing the averages and moments of...

This chapter is devoted to the analysis of sequential measurements of physical quantities, in which the “time” variable may be either continuous or discrete. Formal definitions and classifications of random variables and random processes precede the important topics of the (generalized) central limit theorem and stable distributions, both in the co...

The ability to reliably solve initial-value problems for ordinary differential equations is essential in order to understand the evolution of dynamical systems. In this chapter we deal with methods of advancing the given initial state of a system to later times, explaining clearly the role of stiffness, local discretization and round-off errors, an...

Finite-difference methods for partial differential equations in several dimensions are presented by first handling the basic (parabolic) diffusion equation in two dimensions by explicit and implicit difference schemes, allowing us to introduce the corresponding stability criteria. The treatment of alternating direction implicit (ADI) schemes is sup...

The representation of spatial derivatives is at the heart of spectral methods for partial differential equations, so the three main kinds (Fourier, Chebyshev and Legendre) are analyzed at the outset, together with efficient means to compute them. Galerkin methods involving all three classes of basis functions are discussed for both stationary (Helm...

After reviewing the basic properties of floating-point representation, examples of typical use of expressions in finite-precision arithmetic are given, along with illustrations of common programming pitfalls. Selected classes of function approximation are shown next: optimal (minimax) and rational (Padé) approximations, as well as approximations of...

Solving non-linear equations and systems of such equations is one of the daily chores of a physicist or engineer. In this chapter basic techniques for scalar equations are explained: bisection, Newton–Raphson with optional convergence improvements, the secant and Müller’s method. Vector non-linear equations are treated by Newton–Raphson and Broyden...

Solving systems of linear equations, linear least-square problems and matrix eigenvalue problems is handled by a myriad of freely available and commercial tools. Yet even the most basic operations like the multiplication of two matrices or computing a determinant can be excessively time-consuming and inaccurate if performed recklessly. This chapter...

We present a simple kinematic model of a non-equilibrium steady state device, which can operate either as a heat engine or as a refrigerator. The model is composed of two or more scattering channels where the motion is fully described by deterministic classical dynamics, which connect a pair of stochastic (infinite) heat and particle baths at unequ...

We propose a simple classical dynamical model of a thermoelectric (or thermochemical) heat engine based on a pair of ideal gas containers connected by two unequal scattering channels. The model is solved analytically and it is shown that a suitable combination of parameters can be chosen such that the engine operates at Carnot's efficiency.

We present a numerical method for calculation of Ruelle-Pollicott resonances of dynamical systems. It constructs an effective coarse-grained propagator by considering the correlations of multiple observables over multiple timesteps. The method is compared to the usual approaches on the example of the perturbed cat map and is shown to be numerically...

The ensemble of random Markov matrices is introduced as a set of Markov or stochastic matrices with the maximal Shannon entropy. The statistical properties of the stationary distribution pi, the average entropy growth rate $h$ and the second largest eigenvalue nu across the ensemble are studied. It is shown and heuristically proven that the entropy...

The triangle map on the torus is a non-hyperbolic system featuring properties mainly common to chaotic systems such as diffusion, ergodicity and mixing. Here we present some new aspects of ergodicity and mixing in this system. The properties of the triangle map are studied by symbolically encoding the evolution up to some time t using two different...

In this paper we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters defining the map. The dynamics is studied numerically by means of two different symbolic encoding schemes, both rel...

Linear chains of quantum scatterers are studied in the process of lengthening, which is treated and analysed as a discrete dynamical system defined over the manifold of scattering matrices. Elementary properties of such dynamics relate the transport through the chain to the spectral properties of individual scatterers. For a single-scattering chann...

We study the time evolution of the quantum-classical correspondence (QCC) for the well-known model of quantized perturbed cat maps on the torus in the very specific regime of semi-classically small perturbations. The quality of the QCC is measured by the overlap of the classical phase-space density and corresponding Wigner function of the quantum s...

A detailed analysis of the wave-mode structure in a bend and its incorporation into a stable algorithm for calculation of the scattering matrix of the bend is presented. The calculations are based on the modal approach. The stability and precision of the algorithm is numerically and analytically analysed. The algorithm enables precise numerical cal...

A spatially extended two-dimensional billiard system named serpent billiard is studied in quantum and classical picture. It is composed of bends and straight wave-guide segments forming a channel (billiard chain) with smooth and parallel walls. The billiard possesses a property called the uni-directional transport by which the classical particles t...

We study Wigner function value statistics of classically chaotic quantum maps on compact 2D phase space. We show that the Wigner function statistics of a random state is a Gaussian, with the mean value becoming negligible compared to the width in the semi-classical limit. Using numerical example of quantized sawtooth map we demonstrate that the rel...

We have studied statistical properties of the values of the Wigner function W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this purpose we have defined a Wigner function probability distribution P(w) = (1/V) int delta(w-W(x)) dx, which has, by definition, fixed first and second moment. In particular, we concentrate on relax...

We propose to study the $L^2$-norm distance between classical and quantum phase space distributions, where for the latter we choose the Wigner function, as a global phase space indicator of quantum-classical correspondence. For example, this quantity should provide a key to understand the correspondence between quantum and classical Loschmidt echoe...

We prove an analytical expression for the size of the gap between the ground and the first excited state of quantum adiabatic algorithm for the 3-satisfiability, where the initial Hamiltonian is a projector on the subspace complementary to the ground state. For large problem sizes the gap decreases exponentially and as a consequence the required ru...

We revisit the problem of the dynamical quantum-classical correspondence in classically chaotic systems for pure quantum states. We study time evolution of the deviation between the Wigner function of a quantum state and the corresponding classical density distribution by computing its overlap $F(t)$ -- {\it classical quantum fidelity}. As for the...

We present a dynamical analysis of a classical billiard chain-a channel with parallel semi-circular walls, which can serve as a prototype for a bended optical fiber. An interesting feature of this model is the fact that the phase space separates into two disjoint invariant components corresponding to the left and right uni-directional motions. Dyna...

この論文は国立情報学研究所の電子図書館事業により電子化されました。 We outline an analysis of a classical billiard chain - a channel that can serve as a model for bended optical fibers. The phase space is split into two dis-joined parts corresponding to left and right unidirectional motion. The dynamics is analyzed in terms of a jump model defined by a jump map and a time function....

We present a dynamical analysis of a classical billiard chain---a channel with parallel semi-circular walls, which can serve as a model for a bent optical fibre. An interesting feature of this model is the fact that the phase space separates into two disjoint invariant components corresponding to the left and right uni-directional motions. Dynamics...

We studied statistical properties of Wigner functions $W(x)$ of 1D quantum maps on compact phase space of finite area $V$. For this purpose we defined a Wigner function probability distribution $P(w) = 1/V \int \delta (w-W(x)) dx$, which has, by definition, fixed first and second moment. In particular, we concentrate on relaxation of time evolving...