abstract

Noise-free aperiodic stochastic multiresonance in a chaotic map — a classical kicked spin model with damping — close to the attractor merging crisis is investigated. The input aperiodic signal, in a form of Gaussian correlated noise, is superimposed on the control parameter (the strength of the magnetic field pulses), and the output signal reflects jumps between two symmetric parts of the attractor above the crisis point. As the internal chaotic dynamics is varied by increasing the mean value of the control parameter, multiple maxima of the input-output correlation function are observed. This is due to the fractal structure of the precritical attractors and their basins of attraction colliding at the crisis point. The numerical results are confirmed by analytic evaluation of the correlation function, based on simple models of the colliding fractal sets. The observed phenomenon bears much resemblance to noise-free stochastic multiresonance with periodic signal observed in the same model, but the multiple maxima of the correlation function are less distinct due to the long tails in the probability distribution of the aperiodic signal.

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Noise-free stochastic resonance is demonstrated numerically in a model for Rayleigh–Bénard turbulence in a spatially extended system, based on a one-dimensional array of coupled chaotic Lorenz cells. The system shows spatiotemporal intermittency as the control parameter — equivalent to the temperature difference between the upper and lower surface of the liquid layer — is increased. If the temperature difference is slowly modulated periodically, the signal-to-noise ratio, obtained from the output signal reflecting the occurrence of laminar and turbulent phases in a given point in space, shows maximum as a function of the mean value of the control parameter. The results suggest that experimental observation of noise-free stochastic resonance in spatially extended systems is possible.

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We investigate the band velocity \(v\) and the diffusion coefficient \(D\) of DNA in gel for the geometration mechanism. Here we treat the geometration as a one-step process. The velocity dispersion shows a maximum for some small number of hookings. This allows to expect that the diffusion constant behaves in a similar way. On the other hand, the average number of hookings should increase with the molecule length. Summarizing, the diffusion constant should decrease for very long molecules. However, we do not find this effect in experimental data. This contradiction can be resolved by a conclusion that multiple hookings do not occur.

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The cluster structure of Toom North-East-Center (NEC) voting rule in probabilistic cellular automata stationary states is analyzed. Such structure has its origin in both geometrical connectivity and Toom interactions. The difference between percolation threshold and ferromagnetic phase transition is determined. The value of this difference depends on the way in which NEC rule is applied: synchronous or asynchronous.

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We analyze a specific role of probability density gradients in the theory of irreversible transport processes. The classic Fisher information and information entropy production concepts are found to be intrinsically entangled with the very notion of the Markovian diffusion process and that of the related (local) momentum conservation law.

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Recently, the strong friction limit for a quantum system coupled to a heat bath environment has been explored starting from the exact path integral formulation. Generalizing the classical Smoluchowski limit to low temperatures a time evolution equation for the position distribution, the quantum Smoluchowski equation, has been derived. This important result can even be extended to a quantum Fokker–Planck equation in full phase space. Here, we review these fundamental findings from a physical perspective and apply them to the Kramers barrier escape problem at low temperatures and strong friction.

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Some aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov–Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. A special attention is devoted to finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system.

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We analyze a thermally activated bimolecular reaction in a dilute gas proceeding with introduction of the Prigogine–Xhrouet model (PX) for the reactive cross section. We use the Shizgal–Karplus perturbation method of solution of the Boltzmann equation for reactions \(A + A \rightarrow B + B\) and \(A + A \rightleftarrows B + B\) to obtain the analytical expressions for the nonequilibrium temperatures of reagents and for the rate of chemical reaction. We present the results obtained within one and two Sonine polynomials approximations. The rate constant of chemical reaction depends on concentration of products for the first reaction only and for the second reaction is constant. The analytical results for the temperature of the reagent \(A\) and its value in the beginning of reaction for the product \(B\) are compared with those obtained from the Monte Carlo computer simulations with use of the Bird method. It is shown that the nonequilibrium effects in Shizgal–Karplus temperatures and in decrease of the chemical constant rate are more pronounced than for the lines-of-centers model. For the PX model the rate constant can be decreased even 4 times.

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The functional integral representation of quantum statistical mechanics by means of the Feynman–Kac formula leads to a classical-like description of the system. Point quantum particles are then described in terms of random loops (closed Brownian paths), and all techniques of classical statistical mechanics become available. One advantage of this formalism is that it is not perturbative with respect to the interaction strength, in contrast to the standard many-body perturbative treatment. We apply these ideas to the Coulomb gas by constructing an effective potential (the quantum analogue of the Debye potential) that incorporates both long distance collective screening effects as well as the short range quantum mechanical binding. For Bose systems, we show that mean field theory corresponds to summing all tree-graphs and investigate how to go beyond the mean field description.

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We study anomalous relaxation properties of the continuous-time random walk model in which the space-jump and waiting-time evolution is given by two random Markov processes. This model describes the subordination of one random process by another. The directing process is inverse to the totally skewed, strictly Lévy process. Owing to the properties of the directing process, the relaxation function in the uncoupled random walk model takes the empirical Cole–Cole form. By means of this theoretical analysis we find that the coupled and uncoupled walks lead to different forms of the relaxation function.

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We review some results that illustrate the constructive role of noise in nonlinear systems. Several phenomena are briefly discussed: optimal localization of orbits in a system with limit cycle behavior and perturbed by colored noise; stochastic branch selection at secondary bifurcations; noise-induced order/disorder transitions and pattern formation in spatially extended systems. In all cases the presence of noise is crucial, and the results reinforce the modern view of the importance of noise in the evolution of nonlinear systems.

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Partitioned iterated function system-semifractals (PIFS-SF) and generalized fractal dimension (\(D_q\)) or \(f(\alpha )\) formalism to analysis of amacrine dendrites of gold fish retina has been applied. The number of codes (PIFS-SF) has been used as a measure of dendrites complexity in comparison with \(D_q/f(\alpha )\) behaviour. Some structure-functions (electrophysiological) correlations based on \(D_q\) negative part have also been shown.

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We present experimental results of the anomalous diffusion in a membrane system obtained by means of the laser interferometric technique. We show that in membrane system with glucose diffusing in a gel medium, the thickness of the concentration boundary layers manifest a subdiffusive character. Namely, the thickness of the concentration boundary layer \(\delta \) scales as \(t^{\gamma }\) with \(\gamma =0.443\pm 0.019\).

abstract

The subdiffusion is defined by the relation \(\langle (\Delta x)^{2}\rangle \sim t^{\alpha }\) with \(\alpha \lt 1\). We present a new method of calculating the coefficient \(\alpha \), which can be used to obtain it experimentally. The method refers to the time evolution of the thickness of the so-called near-membrane layers.

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A method is described for the assessment of the complexity of short data sets by nonlinear dynamics. The method was devised for and tested on human heart rate recordings approximately 2000 to 9000 RR intervals long which were extracted from the memory of implantable defibrillator devices (ICD). It is, however, applicable in a more general context. The ICDs are meant to control life-threatening episodes of ventricular tachycardia and/or ventricular fibrillation by applying a electric shock to the heart through intracardiac electrodes. It is well known that conventional ICD algorithms yield approximately 20–30% of spurious interventions. The main aim of this work is to look for nonlinear dynamics methods to enhance the appropriateness of the ICD intervention. We first showed that nonlinear dynamics methods first applied to 24-hour heart rate variability analysis were able to detect the need for the ICD intervention. To be applicable to future ICD use, the methods must also be low in computational requirements. Methods to analyse the complexity of the short and non-stationary sets were devised. We calculated the Shannon entropy of symbolic words obtained in a sliding 50 beat window and analysed the dependence of this complexity measure on the time. Precursors were found extending much earlier time than the time the standard ICD algorithms span.

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Two different frameworks of the relaxation phenomenon are considered to conduct a detailed analytical analysis on the relationship of the stretched exponential relaxation function and the relaxation function resulting from a certain log-relaxation-time density proposed by Rajagopal. In the first part the analytical comparison of these two functions in the purely heterogeneous picture is presented. The considerations are based on the interpretation of the relaxation function as a survival probability of the initial state of a relaxing system expressed by means of the weighted average of an exponential decay with respect to the distribution of the effective relaxation time. In the second part a certain degree of intrinsic nonexponentiality is assumed which allows to show the stochastic scheme leading directly from the Rajagopal density to the stretched exponential relaxation response. In both approaches the strict connection of Rajagopal function and the one-sided stable density is shown.

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The evolution of quantum correlations for \(XXZ\) model is studied. It is shown that the simplified entanglement measure which follows from entanglement of formation can be used as an efficient tool in investigating the properties of the dynamics in question. In particular, the behavior of this measure for pure states gives us information about decoherence or entanglement that can occur during the time evolution for the system. We present some numerical results which confirm that the generalized conditional expectation defining the stochastic dynamics for \(XXZ\) model contains the proper (i.e. genuine quantum) interactions between subsystem and its environment.

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Besides three-dimensional autonomous nonlinear dynamical systems, periodically driven nonlinear oscillators constitute elementary classes of systems that can potentially exhibit chaotic behavior. In this contribution we investigate conditions on the shape of the potential and the functional form of the periodic driving that are necessary for the occurrence of chaotic behavior in these systems by deriving analytical criteria that exclude chaotic long-time solutions.

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We quantify the concept of ensembles of atoms’ fluctuations for 2D liquids close to and in the two-phase coexistence region, using a probabilistic method of local structure analysis. Two ensembles are studied: gas-like independent gaussian fluctuations (IGF) A.C. Mituś, A.Z. Patashinski, A. Patrykiejew, S. Sokolowski, Phys. Rev. B66, 184202 (2002) and a truly solid-like fluctuations (SF) characteristic for a two-dimensional (2D) triangular solid. For a 2D Lennard–Jones (LJ) system simulated using Monte Carlo and molecular dynamics methods those ensembles yield a statistically reliable description of local solid-like structures in the two-phase region. Both ensembles undergo a spectacular breakdown as the density changes. A hypothetical relation of this breakdown to the behaviour of the heat capacity in the two-phase region is proposed.

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Arrhythmia is a condition in which an additional ectopic pacemaker is present in the tissue of the heart. Localization of ectopic foci is essential for successful radio-frequency ablation, an important surgical way of treating arrhythmia. In one of the possible mechanisms, arrhythmia induced by an ectopic foci located in one of the main blood vessels leading out or onto the heart. The therapeutic procedure in this case is usually ablation of the whole junction of the blood vessel with heart wall. In this way, whatever excitation occurs inside the vessel, it cannot penetrate the ventricles perturbing their contraction cycle. Such an ablation procedure is long and burdened with the risk of the perforation. A more safe method would involve the localization of the source of the excitation (i.e. the ectopic foci) and its ablation. The methods used in cardiology at present involve complicated localization systems and are time-consuming with the patient spending a long time on the operating table. Recently, Hall and Glass have developed numerical methods which allow to quickly to model the localization of the ectopic foci in a flat, square sample of an inhomogeneous medium. Here, we demonstrate an extension of this model for the case of a cylinder containing an ectopic foci, that can be a model of a blood vessel with the source of the ectopic beat inside it. Three methods of localization are implemented. Standard electrodes containing several active tips are used to stimulate the medium locally and locate the foci judging from the reaction of the system. The first one uses electrode activation times to compute the location of the ectopic site. The second one localizes it by measuring the resetting response of the foci, and the third one, uses wavefront curvature. Specifically for the cylindrical geometry of the blood vessel, we developed a localization procedure that allows to quickly localize the pacemaker.

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We show that the most popular estimators of the self-similarity index — the Hurst and the DFA exponents — cannot give exact value of the estimated parameter in some cases. The goal of this paper is to provide a simple computer test by means of which origins of the self-similarity feature of a particular time series can be found. We demonstrate that the observed self-similarity can reflect a long-memory (fractional Brownian motion case) or infinite variance of the process’ increments (Lévy \(\alpha \)-stable motion case).

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We study magnetic fluxes and currents in a set of mesoscopic rings which form a cylinder. We investigate the noiseless system as well as the influence of equilibrium and non-equilibrium fluctuations on the properties of selfsustaining currents. Thermal equilibrium Nyquist noise does not destroy selfsustaining currents up to temperatures of the same order as the critical temperature for selfsustaining currents. For temperatures below the critical temperature, randomness in the distribution of parity of the coherent electrons can lead to disappearing of selfsustaining currents and inducing new metastable states. For temperatures above the critical temperature, it causes a creation of new metastable states with non-zero currents.

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We study short period changes in galactic cosmic rays (GCR) intensity and analyze features of its sporadic and recurrent decreases (Forbush effects). We show that the energy spectrum of Forbush effect is suitable for investigating the structure of irregularities of the interplanetary magnetic field, resulting from shock waves and magnetic clouds in the interplanetary space. This study is based on experimental data from neutron monitors and on theoretical modeling of GCR transport.