Numerical investigation of self-organizing systems that arise in plastic information

Solutions of the equations of the evolution of the most popular self-organizing systems observed when a solid is deformed are investigated. The phase patterns for a “predator—prey” self-excited oscillatory system which arises during the evolution of a dislocation—disclination type pair, an autocatalytic system, formed by dislocation—vacancy type defects in a localized zone of plastic shear, and the self-organizing system described by the field of elastic stresses and strains for a solid with wave-type plastic deformation are described. Sumskii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 3–8, June, 1996.     

Comments on the information,information gain and efficiency of self-organizing systems

Haken recently applied the slaving principle to decompose expressions for information, information gain and efficiency of self-organizing systems into two parts. The first part in each case depends on the order parameter while the second on the slaved modes. Close to the instability points the latter contribution can be ignored. In this paper we elaborate on the formulas given by Haken and find exact expressions which are analytical in the entire domain of the values of control parameters. We also find their asymptotic limits in the immediate proximity of the instability point and far away from it. Our results provide a direct way of relating information, information gain and efficiency of nonequilibrium critical systems to their control parameters.     

Divergence and Shannon information in genomes

Shannon information (SI) and its special case, divergence, are defined for a DNA sequence in terms of probabilities of chemical words in the sequence and are computed for a set of complete genomes highly diverse in length and composition. We find the following: SI (but not divergence) is inversely proportional to sequence length for a random sequence but is length independent for genomes; the genomic SI is always greater and, for shorter words and longer sequences, hundreds to thousands times greater than the SI in a random sequence whose length and composition match those of the genome; genomic SIs appear to have word-length dependent universal values. The universality is inferred to be an evolution footprint of a universal mode for genome growth.     

Shannon information entropies for rectangular multiple quantum well systems with constant total lengths

We first study the Shannon information entropies of constant total length multiple quantum well systems and then explore the effects of the number of wells and confining potential depth on position and momentum information entropy density as well as the corresponding Shannon entropy.We find that for small full width at half maximum(FWHM) of the position entropy density,the FWHM of the momentum entropy density is large and vice versa.By increasing the confined potential depth,the FWHM of the position entropy density decreases while the FWHM of the momentum entropy density increases.By increasing the potential depth,the frequency of the position entropy density oscillation within the quantum barrier decreases while that of the position entropy density oscillation within the quantum well increases.By increasing the number of wells,the frequency of the position entropy density oscillation decreases inside the barriers while it increases inside the quantum well.As an example,we might localize the ground state as well as the position entropy densities of the1 st,2 nd,and 6 th excited states for a four-well quantum system.Also,we verify the Bialynicki–Birula–Mycieslki(BBM)inequality.     

Local wave-vector, Shannon and Fisher information

It is shown that the local wave-vector gives the connection between the Shannon and Fisher information: the local wave-vector is the gradient of the Shannon information per particle and the square of the local wave-vector is the Fisher information per particle.     

Physical information entropy and probability Shannon entropy

In a previous work by one of us (R. Urigu) concerning open quantum systems it was remarked that in processes of the type , when evaluating the information entropy of the environment as the Shannon entropy of the outcome probabilities in the channels , the total information entropy may decrease. We remark here that this decrease is easily excluded by requiring a condition of quantum modelizability of the environment even with respect to Shannon entropy (“cybernetic interpretability” of the environment). Further conditions on the quantum model of the environment are defined (“maximal observability” and “Boolean interpretability”), which are proved to be equivalent, and it turns out that, once satisfied in one model, they also are in any model with pure initial state; furthermore, these conditions turn out to be equivalent to the condition that the process consists of pure operations of the first kind. The relevance to the concept of macroscopicity and to the “von Neumann chain” is discussed.     

Information,information gain,and efficiency of self-organizing systems close to instability points

Selforganizing systems are those which can acquire macroscopic spatial, temporal, or spatio-temporal structures without interference from the outside by mere changes of control parameters. The slaving principle which has been derived for such systems close to their instability points allows one to give the probability distribution of the total system a specific form.This in turn allows one to decompose information and information gain into a part which refers to the order parameters alone and a second part which is a sum over the information of the slaved modes averaged over the distribution of the order parameters. Close to instability points the information of the order parameters changes dramatically whereas the information of the slaved modes does not. Therefore close to these points it is sufficient to study the behavior of the order parameter information and information gain which is done explicitly for a large class of systems undergoing nonequilibrium phase transitions. It is shown how information and information gain as well as efficiency (in the sense defined in this paper) can be measured directly.     

A comparison of the Shannon and Kullback information measures

Two widely used information measures are compared. It is shown that the Kullback measure, unlike the Shannon measure, provides the basis for a consistent theory of information which extends to continuous sample spaces and to nonconstant prior distributions. It is shown that the Kullback measure is a generalization of the Shannon measure, and that the Kullback measure has more reasonable additivity properties than does the Shannon measure. The results lend support to Jaynes's entropy maximization procedure.     

Estimating the Shannon entropy: recurrence plots versus symbolic dynamics

Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy is solved here by using a new definition--still based on the recurrence plots--and it is verified that this new definition is correlated with the largest Lyapunov exponent, as expected from the Pesin conjecture. A comparison with a Shannon entropy computed from symbolic dynamics is also provided.     

Differences and similarities between self-organizing and organized systems

The energy and dynamic characteristics of self-organizing and organized physical systems in which particles interact via electric fields are consistently considered in terms of the integral Lagrangian minimization. It is shown that the energy dissipation plays a minor role in self-organization processes. Two types of work are distinguished, basic differences and similarities between the systems are revealed, and comparison with living organisms is made.     

The Fisher information measure and Shannon entropy for particulate matter measurements

We investigated the dynamics of particulate matter data, recorded in Tito, a small industrial area of southern Italy. The analysis of these signals was performed using the Fisher information measure (FIM), which is a powerful tool for investigating complex and nonstationary signals, and the Shannon entropy, which is a well-known tool for investigating the degree of disorder in dynamical systems. Our results point to an increase of disorder and complexity from fine to coarse particulates.     

Critical conditions for spontaneous relaxation in self-organizing dissipative film systems

It is established that the necessary conditions for spontaneous relaxation of elastic strain energy in a copper-As₆₀Se₄₀ self-organizing dissipative heterostructure is that the elastic deformation energy and the temperature must reach their threshold values. It is shown that in the temperature range 270–340 K the spontaneous relaxation of elastic deformation energy is accompanied by structural-chemical ordering and anomalous diffusion of copper into the glassy chalcogenide semiconductor layer. The maximum concentration of copper dissolved in the films is 40 at. %. Conductivity inversion from p to n type is observed in doped layers obtained by this method. Zh. Tekh. Fiz. 69, 128–129 (August 1999)     

Shannon entropies and Fisher information of K-shell electrons of neutral atoms

We represent the two K-shell electrons of neutral atoms by Hylleraas-type wave function which fulfils the exact behavior at the electron–electron and electron-nucleus coalescence points and, derive a simple method to construct expressions for single-particle position- and momentum-space charge densities, $\rho (\mathit{r})$ and $\gamma (\mathit{p})$ respectively. We make use of the results for $\rho (\mathit{r})$ and $\gamma (\mathit{p})$ to critically examine the effect of correlation on bare (uncorrelated) values of Shannon information entropies (S) and of Fisher information (F) for the K-shell electrons of atoms from helium to neon. Due to inter-electronic repulsion the values of the uncorrelated Shannon position-space entropies are augmented while those of the momentum-space entropies are reduced. The corresponding Fisher information are found to exhibit opposite behavior in respect of this. Attempts are made to provide some plausible explanation for the observed response of S and F to electronic correlation.     

Local coordinate,wave vector,Fisher and Shannon information in momentum representation

The formalism of Luo [Int. J. Theor. Phys. 41 (2002) 1713] on local values of quantum observables is generalized for N-electron systems both in coordinate and momentum spaces. It is shown that the imaginary part of the total local coordinate (momentum) is the half of the local wave vector (or the half of the gradient of the local momentum(coordinate)-space Shannon information per particle).     

Oscillatory Shannon entropy in the process of equilibration of nonequilibrium crystalline systems

We present a study of the equilibration process of some nonequilibrium crystalline systems by means of molecular dynamics simulation technique. The nonequilibrium conditions are achieved in the systems by randomly defining velocity components of the constituent atoms. The calculated Shannon entropy from the probability distribution of the kinetic energy among the atoms at different instants during the process of equilibration shows oscillation as the system relaxes towards equilibrium. Fourier transformations of these oscillating Shannon entropies reveal the existence of Debye frequency of the concerned system.     

Comparison of the Shannon information, Tsallis information, and number of system states in diagnosing regular and chaotic motions

The properties of the Shannon and Tsallis informations in diagnosing laminar and turbulent motions are compared by solving the set of Lorentz equations. A method for analyzing the mode of dynamic behavior by constructing the function of number of states is proposed.     

Evolution of self-organizing systems from the standpoint of mechanics and thermodynamics

The extension of the principle of least action for many-particle systems has shown that the natural evolution of processes in the environment is governed largely by kinetic-to-potential energy conversion. A relation with the variational principles of nonequilibrium thermodynamics is established. It is shown that the birth and aging of structures are controlled by the principle of minimization of the integral Lagrangian, the role of which in mechanics is not less significant than the role of the law of degradation of energy (the second law of thermodynamics) in thermodynamics.     

Entanglement versus correlations in spin systems

We consider pure quantum states of N>1 spins or qubits and study the average entanglement that can be localized between two separated spins by performing local measurements on the other individual spins. We show that all classical correlation functions provide lower bounds to this localizable entanglement, which follows from the observation that classical correlations can always be increased by doing appropriate local measurements on the other qubits. We analyze the localizable entanglement in familiar spin systems and illustrate the results on the hand of the Ising spin model, in which we observe characteristic features for a quantum phase transition such as a diverging entanglement length.