Dynamical Entropy Research Articles

Study of entropy-diffusion relation in deterministic Hamiltonian systems through microscopic analysis.

Although an intimate relation between entropy and diffusion has been advocated for many years and even seems to have been verified in theory and experiments, a quantitatively reliable study and any derivation of an algebraic relation between the two do not seem to exist. Here, we explore the nature of this entropy-diffusion relation in three deterministic systems where an accurate estimate of both can be carried out. We study three deterministic model systems: (a) the motion of a single point particle with constant energy in a two-dimensional periodic potential energy landscape, (b) the same in the regular Lorentz gas where a point particle with constant energy moves between collisions with hard disk scatterers, and (c) the motion of a point particle among the boxes with small apertures. These models exhibit diffusive motion in the limit where ergodicity is shown to exist. We estimate the self-diffusion coefficient of the particle by employing computer simulations and entropy by quadrature methods using Boltzmann's formula. We observe an interesting crossover in the diffusion-entropy relation in some specific regions, which is attributed to the emergence of correlated returns. The crossover could herald a breakdown of the Rosenfeld-like exponential scaling between the two, as observed at low temperatures. Later, we modify the exponential relation to account for the correlated motions and present a detailed analysis of the dynamical entropy obtained via the Lyapunov exponent, which is rather an important quantity in the study of deterministic systems.

Investigation of denoising effects on forecasting models by statistical and nonlinear dynamic analysis

Abstract In this study, the denoising effect on the performance of prediction models is evaluated. The 13-year daily data (2002–2015) of hydrological time series for the sub-basin of Parishan Lake of the Helle Basin in Iran were used to predict time series. At first, based on observational precipitation and temperature data, the prediction was performed, using the ARIMA, ANN-MLP, RBF, QES, and GP prediction models (the first scenario). Next, time series were denoised using the wavelet transform method, and then the prediction was made based on the denoised time series (the second scenario). To investigate the performance of the models in the first and second scenarios, nonlinear dynamic and statistical analysis, as well as chaos theory, was used. Finally, the analysis results of the second scenario were compared with those of the first scenario. The comparison revealed that denoising had a positive impact on the performance of all the models. However, it had the least influence on the GP model. In the time series produced by all the models, the error rate, embedding dimension needed to describe the attractors in dynamical systems and entropy decreased, and the correlation and autocorrelation increased.

Wehrl entropy production rate across a dynamical quantum phase transition

This paper proposes the Husimi-Q function based entropy as a suitable quantity to describe the thermodynamic properties of dynamical phase transitions, and establishes a relation between dynamical criticality and entropy production.

Classical dynamical coarse-grained entropy and comparison with the quantum version.

We develop the framework of classical observational entropy, which is a mathematically rigorous and precise framework for nonequilibrium thermodynamics, explicitly defined in terms of a set of observables. Observational entropy can be seen as a generalization of Boltzmann entropy to systems with indeterminate initial conditions, and it describes the knowledge achievable about the system by a macroscopic observer with limited measurement capabilities; it becomes Gibbs entropy in the limit of perfectly fine-grained measurements. This quantity, while previously mentioned in the literature, has been investigated in detail only in the quantum case. We describe this framework reasonably pedagogically, then show that in this framework, certain choices of coarse-graining lead to an entropy that is well-defined out of equilibrium, additive on independent systems, and that grows toward thermodynamic entropy as the system reaches equilibrium, even for systems that are genuinely isolated. Choosing certain macroscopic regions, this dynamical thermodynamic entropy measures how close these regions are to thermal equilibrium. We also show that in the given formalism, the correspondence between classical entropy (defined on classical phase space) and quantum entropy (defined on Hilbert space) becomes surprisingly direct and transparent, while manifesting differences stemming from noncommutativity of coarse-grainings and from nonexistence of a direct classical analog of quantum energy eigenstates.

On C*-algebras from homoclinic equivalences on subshifts of finite type

Let G be an infinite countable group and A be a finite set. If ∑ ⊆ AG is a strongly irreducible subshift of finite type, we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation ${\cal G}$ on ∑ and show that the reduced C*-algebra $C_r^*\left({\cal G} \right)$ of ${\cal G}$ is a unital simple approximately finite (AF)-dimensional C*-algebra. The shift action of G on ∑ induces a canonical automorphism action of G on the C*-algebra $C_r^*\left({\cal G} \right)$ . We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C*-algebras, and show that, if G is an amenable group, then the noncommutative topological entropy of the canonical automorphism action of G on $C_r^*\left({\cal G} \right)$ is equal to the topology entropy of the shift action of G on ∑. We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C*-dynamical system ( $C_r^*\left({\cal G} \right)$ , G).

Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata

In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$.

Application of a Diagnostic Methodology of Cardiac Systems Based on the Proportions of Entropy in Normal Patients and with Different Pathologies

Introduction: Dynamical systems theory, probability, and entropy were the substrate for the development of the diagnostic and predictive methodology of adult heart dynamics. Objective: To apply a previously developed methodology from dynamical systems, probability, and entropy in both normal and pathological subjects. Methods: Electrocardiographic records were selected from 30 healthy subjects and 200 with different pathologies, with a length of least 18 h. Numerical attractors from dynamical attractors and the probability of occurrence of ordered pairs of consecutive heart rates were built. A calculation of entropy and its proportions was performed and statistical analysis was conducted. Results: The normal patients’ heart dynamics were evaluated according to the methodology of entropy proportions, highlighting that there are differences in normal patients with different pathologies. There was maximal level of sensitivity, specificity, and diagnostic agreement. Conclusion: Proportional entropy constitutes a diagnostic and predictive method of heart systems, and may be useful as a tool to objectively diagnose and perform the follow-up of normal and pathological cases.

On transmission efficiency of quantum modulations

In quantum information theory, it is important to find modulations with low information loss for noisy channels. In this paper, using the quantum dynamical entropy and the quantum dynamical mutual entropy, we investigate the transmission efficiency of two quantum modulators through attenuation channels.

Alterations in heart-brain interactions under mild stress during a cognitive task are reflected in entropy of heart rate dynamics

Many people experience mild stress in modern society which raises the need for an improved understanding of psychophysiological responses to stressors. Heart rate variability (HRV) may be associated with a flexible network of intricate neural structures which are dynamically organized to cope with diverse challenges. HRV was obtained in thirty-three healthy participants performing a cognitive task both with and without added stressors. Markers of neural autonomic control and neurovisceral complexity (entropy) were computed from HRV time series. Based on individual anxiety responses to the experimental stressors, two subgroups were identified: anxiety responders and non-responders. While both vagal and entropy markers rose during the cognitive task alone in both subgroups, only entropy decreased when stressors were added and exclusively in anxiety responders. We conclude that entropy may be a promising marker of cognitive tasks and acute mild stress. It brings out a new central question: why is entropy the only marker affected by mild stress? Based on the neurovisceral integration model, we hypothesized that neurophysiological complexity may be altered by mild stress, which is reflected in entropy of the cardiac output signal. The putative role of the amygdala during mild stress, in modulating the complexity of a coordinated neural network linking brain to heart, is discussed.

Comparison of selected measures of gait stability derived from center of pressure displacement signal during single and dual-task treadmill walking

Steady state gait dynamics has been examined using the measures of regularity, local dynamic stability, and variability. This study investigates the relationship between these measures under increasing cognitive loads. Participants walked on an instrumented treadmill at 1 m/s under walk only and two dual-task conditions. The secondary tasks were visuomotor cognitive games (VCG) of increasing difficulty level. The center of pressure displacement in the mediolateral direction (ML COP-D) and cognitive game performance were recorded for analysis. The following measures were calculated: (1) sample entropy (SampEn) and quantized dynamical entropy (QDE) of the ML COP-D, (2) short-term largest Lyapunov exponent (LLE) of the ML COP-D, and (3) variability of inter-stride spatio-temporal gait variables. Entropy and variability measures significantly increased from walk only to both dual-task conditions. Whereas, the short-term LLE increased only during the easy VCG task. No measure was sensitive to the difficulty level of the VCG tasks. The variability of heel strike positions in the mediolateral direction was positively correlated with SampEn and QDE. However, there were no significant correlations between the short-term LLE and either variability measures or entropy measures. These findings confirm that each of these measures is representative of a different aspect of human gait dynamics.

Optimality in quantum data compression using dynamical entropy

In this article we study lossless compression of strings of pure quantum states of indeterminate-length quantum codes which were introduced by Schumacher and Westmoreland. Past work has assumed that the strings of quantum data are prepared to be encoded in an independent and identically distributed way. We discuss the notion of quantum stochastic ensembles, allowing us to consider strings of quantum states prepared in a general way. For any quantum stochastic ensemble we define an associated quantum dynamical system and prove that the optimal average codeword length via lossless coding is equal to the quantum dynamical entropy of the associated quantum dynamical system.

Correlating Melt Dynamics and Configurational Entropy Change With Topological Phases of AsxS100–x Glasses and the Crucial Role of Melt/Glass Homogenization

Melt dynamics and glass Topological phases of especially dry and homogenized binary AsxS100-x melts/glasses are examined in Modulated-DSC, Raman scattering and volumetric experiments. In the S-rich glasses (12% 20 as x increases. Taken together, these results underscore that superstrong melts yield isostatically rigid glasses, while fragile ones form either Flexible or Stressed-rigid glasses upon cooling. The onset of the rigidity transition near = 2.22, instead of the usual value of = 2.40, is identified with presence of QT local structures in addition to Pyramidal As(S1/2)3 local structures in the glassy backbone, and with a small but finite fraction of polymeric Sn chains being decoupled from the backbone.

The Lyapunov spectra of quantum thermalisation

Thermalisation in closed quantum systems occurs through a process of dephasing due to parts of the system outside of the window of observation, gradually revealing the underlying thermal nature of eigenstates. In contrast, closed classical systems thermalize due to dynamical chaos. We demonstrate a deep link between these processes. Projecting quantum dynamics onto variational states using the time-dependent variational principle, results in classical chaotic Hamiltonian dynamics. We study an infinite spin chain in two ways—using the matrix product state ansatz for the wavefunction and for the thermofield purification of the density matrix—and extract the full Lyapunov spectrum of the resulting dynamics. We show that the entanglement growth rate is related to the Kolmogorov–Sinai entropy of dynamics projected onto states with appropriate entanglement, extending previous results about initial entanglement growth to all times. The Lyapunov spectra for thermofield descriptions of thermalizing systems show a remarkable semi-circular distribution.

Quantum Dynamical Mutual Entropy Based on AOW Entropy

The classical dynamical mutual entropy measures the average of information content going through a channel. Classical Markovian sources are important in communication theory since they constitute reasonable models for languages. In this paper, we define the quantum dynamical mutual entropy through quantum Markov chains, and we calculate it for several simple models.

On the nonlinearity of quantum dynamical entropy

Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Slomczynski and Zyczkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Luders-von Neumann instruments and partitions.

Errors, chaos, and the collisionless limit

We simultaneously study the dynamics of the growth of errors and the question of the faithfulness of simulations of $N$-body systems. The errors are quantified through the numerical reversibility of small-$N$ spherical systems, and by comparing fixed-timestep runs with different stepsizes. The errors add randomly, before exponential divergence sets in, with exponentiation rate virtually independent of $N$, but scale saturating as $\sim 1/\sqrt{N}$, in line with theoretical estimates presented. In a third phase, the growth rate is initially driven by multiplicative enhancement of errors, as in the exponential stage. It is then qualitatively different for the phase space variables and mean field conserved quantities (energy and momentum); for the former, the errors grow systematically through phase mixing, for the latter they grow diffusively. For energy, the $N$-variation of the `relaxation time' of error growth follows the $N$-scaling of two-body relaxation. This is also true for angular momentum in the fixed stepsize runs, although the associated error threshold is higher and the relaxation time smaller. Due to shrinking saturation scales, the information loss associated with the exponential instability decreases with $N$ and the dynamical entropy vanishes at any finite resolution as $N \rightarrow \infty$. A distribution function depending on the integrals of motion in the smooth potential is decreasingly affected. In this sense there is convergence to the collisionless limit, despite the persistence of exponential instability on infinitesimal scales. Nevertheless, the slow $N$-variation in its saturation points to the slowness of the convergence.

Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production

<p style='text-indent:20px;'>We study finite state random dynamical systems (RDS) and their induced Markov chains (MC) as stochastic models for complex dynamics. The linear representation of deterministic maps in RDS is a matrix-valued random variable whose expectation corresponds to the transition matrix of the MC. The instantaneous Gibbs entropy, Shannon-Khinchin entropy of a step, and the entropy production rate of the MC are discussed. These three concepts, as key anchoring points in applications of stochastic dynamics, characterize respectively the uncertainties of a system at instant time <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>, the randomness generated in a step in the dynamics, and the dynamical asymmetry with respect to time reversal. The stationary entropy production rate, expressed in terms of the cycle distributions, has found an expression in terms of the probability of the deterministic maps with single attractor in the maximum entropy RDS. For finite RDS with invertible transformations, the non-negative entropy production rate of its MC is bounded above by the Kullback-Leibler divergence of the probability of the deterministic maps with respect to its time-reversal dual probability.

Design and FPGA Implementation of a Pseudorandom Number Generator Based on a Four-Wing Memristive Hyperchaotic System and Bernoulli Map

Random numbers are widely used in the fields of computer, digital signature, secure communication and information security. Especially in recent years, with the large-scale application of smart card and the demand of information security, the demand for high-quality random number generator is increasingly urgent. With the development of the theory of non-linear systems, the design of pseudorandom number generator (PRNG) for chaotic behavior of non-linear systems provides a new theoretical basis and implementation method. This paper presents a PRNG based on a no-equilibrium four-wing memristive hyperchaotic system (FWMHS) and its implementation on Field Programmable Gate Array (FPGA) board. In order to increase the output throughput and the statistical quality of the generated bit sequences, we propose the PRNG design which uses a dual entropy sources architecture with FWMHS and Bernoulli map. Simulation and experimental results verifying the feasibility of the FWMHS are also given. Then, the proposed PRNG system is modeled and simulated on the Vivado 2018.3 platform, and implemented on the Xilinx ZYNQ-XC7Z020 FPGA evaluation board. The maximum operating frequency has been achieved as 135.04 MHz with a speed of 62.5 Mbit/s. Finally, we have experimentally verified that the binary data obtained by this dual entropy sources architecture pass the tests of NIST 800.22, ENT and AIS.31 statistical test suites with XOR function post-processing for a high throughput speed. The security analysis is carried out by means of dynamical degradation, key space, key sensitivity, correlation and information entropy. Statistical tests and security analysis show that it has good pseudorandom characteristics and can be used in chaos-based cryptographic applications at hardware or software implementation.

Evaluation of heart rate dynamics during meditation using Poincaré phase plane symbolic measures

Meditation has long been known to affect human physiology which is mediated through autonomic nervous system. The main objective of this study was to assess dynamic changes in cardiac inter-beat intervals and autonomic nervous system during meditation hypothesising that Poincare plots in conjunction with symbolic dynamics can lead to better understanding of the dynamics of the underlying physiological mechanisms. The Poincare phase plane standard descriptors can measure only linear gross variability of the heart rate time series. Moreover, these descriptors exhibit limitations where the occurrence of nonlinear behaviour may be responsible to distinguish the phase plane patterns. We captured the changes in Poincare phase plane patterns resulting from nonlinear processes, during two meditation techniques, employing two complexity measures, symbolic dynamics entropy and forbidden words, which are appropriate to distinguish between the two states. Findings show that symbolic dynamics entropy, also a marker of parasympathetic activity, reveals that during premeditation state, there is parasympathetic dominance while during meditation state there is sympathetic dominance in both the meditation techniques. We also show the presence of nonlinear deterministic structures in the meditation inter-beat interval time series and appropriateness of the application of our nonlinear approach through surrogate data analysis.

On the Relation between Topological Entropy and Restoration Entropy

In the context of state estimation under communication constraints, several notions of dynamical entropy play a fundamental role, among them: topological entropy and restoration entropy. In this paper, we present a theorem that demonstrates that for most dynamical systems, restoration entropy strictly exceeds topological entropy. This implies that robust estimation policies in general require a higher rate of data transmission than non-robust ones. The proof of our theorem is quite short, but uses sophisticated tools from the theory of smooth dynamical systems.