Nonrelativistic Shannon information entropy for Kratzer potential

The Shannon information entropy is investigated within the nonrelativistic framework. The Kratzer potential is considered as the interaction and the problem is solved in a quasi-exact analytical manner to discuss the ground and first excited states. Some interesting features of the information entropy densities as well as the probability densities are demonstrated.The Bialynicki–Birula–Mycielski inequality is also tested and found to hold for these cases.     

Onicescu information energy in terms of Shannon entropy and Fisher information densities

Following recent studies concerning the use of information theory in electronic structure theory of atomic and molecular systems, an analytical relationship between Onicescu information energy and densities of Shannon entropy and the two forms of the Fisher information has been presented. The established proof must be viewed in the light of the exponentially decaying nature of the asymptotic density of atoms and molecules.     

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.     

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.     

Basic properties of the generalized Boltzmann-Gibbs- Shannon entropy

We consider the most general expression, S(v∣m), for the entropy of an m-continuous probability measure v relative to a σ-finite reference measure m,both defined on an otherwise arbitrary measurable space. The functional v ?:S(v∣m) is examined with regard to range, continuity and behaviour to coarse-graining v, and we study how S(v∣m) changes if both measures are transfered to another measurable space. Some important entropy inequalities are generalized for the functional v ?: S(v∣m), and we show that, even in this general context, Jaynes' principle distinguishes the Gibbsian probability measures as the typical probability measures with maximum entropy.     

Synergetic information versus Shannon information in self-organizing systems

We discuss how synergetic information, i.e. the compressed Shannon information of the order parameters, which is produced by the cooperativity of the system, can be determined experimentally, especially in fluids.     

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.     

The analogues of entropy and of Fisher's information measure in free probability theory,I

Analogues of the entropy and Fisher information measure for random variables in the context of free probability theory are introduced. Monotonicity properties and an analogue of the Cramer-Rao inequality are proved.Dedicated to Huzuhiro ArakiWork supported in part by a Grant from the National Science Foundation     

Shannon entropy as a measure of uncertainty in positions and momenta

I start with a brief report of the topic of entropic uncertainty relations for the position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of a finite number of detectors set to measure the probability distributions in the position and momentum spaces. I derive the uncertainty relation for the sum of the Shannon entropies which generalizes the previous approach by I. Bialynicki-Birula based on an infinite number of detectors (bins).     

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.     

信息熵、玻尔兹曼熵以及克劳修斯熵之间的关系--兼论玻尔兹曼熵和克劳修斯熵是否等价

论述了信息熵、玻尔兹曼熵以及克劳修斯熵之间的关系;由不涉及具体系统的方法从玻尔兹曼关系、信息熵推导出了克劳修斯熵的表达式;指出玻尔兹曼熵与克劳修斯熵不是等价关系,而是玻尔兹曼熵包含克劳修斯熵,信息熵又包含玻尔兹曼熵。     

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.     

Shannon entropy in the research on stationary regimes and the evolution of complexity

The questions of the identification of complex biological systems (complexity) as special self-organizing systems or systems of the third type first defined by W. Weaver in 1948 continue to be of interest. No reports on the evaluation of entropy for systems of the third type were found among the publications currently available to the authors. The present study addresses the parameters of muscle biopotentials recorded using surface interference electromyography and presents the results of calculation of the Shannon entropy, autocorrelation functions, and statistical distribution functions for electromyograms of subjects in different physiological states (rest and tension of muscles). The results do not allow for statistically reliable discrimination between the functional states of muscles. However, the data obtained by calculating electromyogram quasiatttractor parameters and matrices of paired comparisons of electromyogram samples (calculation of the number k of “coinciding” pairs among the electromyogram samples) provide an integral characteristic that allows the identification of substantial differences between the state of rest and the different states of functional activity. Modifications and implementation of new methods in combination with the novel methods of the theory of chaos and self-organization are obviously essential. The stochastic approach paradigm is not applicable to systems of the third type due to continuous and chaotic changes of the parameters of the state vector x(t) of an organism or the contrasting constancy of these parameters (in the case of entropy).     

Extremal entropy for a constrained probability density

The state of extremal entropy for a one-dimensional probability density is considered. This density is constrained by fixed values of the first and second moment. The grandcanonical distribution yields the extremum of the entropy within a certain range of values of the moments. A different type of density corresponds to an extremum of entropy when the moments are outside this range. The shape of this density is approximated with the Ritz variation method. The results are applied to the formation of breathers in the discrete nonlinear Schrödinger equation.     

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.     

Irreversibility, entropy and incomplete information

The open system has been proved to be a system with perfect accessibility represented as a probability space in which is defined a PA-measure. But, the PA-measure is not yet known; consequently, it is difficult to develop the statistical thermodynamics for an irreversible system. Here its integral expression is obtained in order to its use in the statistical thermodynamic analysis of the complex and irreversible systems.     

Effects of Shannon entropy and electric field on polaron in RbCl triangular quantum dot

In this paper, the time evolution of the quantum mechanical state of a polaron is examined using the Pekar type variational method on the condition of the electric-LO-phonon strong-coupling and polar angle in RbCl triangular quantum dot. We obtain the eigenenergies, and the eigenfunctions of the ground state, and the first excited state respectively. This system in a quantum dot can be treated as a two-level quantum system qubit and the numerical calculations are performed.The effects of Shannon entropy and electric field on the polaron in the RbCl triangular quantum dot are also studied.     

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.     

Recurrence plots and Shannon entropy for a dynamical analysis of asynchronisms in noninvasive mechanical ventilation

Recurrence plots were introduced to quantify the recurrence properties of chaotic dynamics. Hereafter, the recurrence quantification analysis was introduced to transform graphical interpretations into statistical analysis. In this spirit, a new definition for the Shannon entropy was recently introduced in order to have a measure correlated with the largest Lyapunov exponent. Recurrence plots and this Shannon entropy are thus used for the analysis of the dynamics underlying patient assisted with a mechanical noninvasive ventilation. The quality of the assistance strongly depends on the quality of the interactions between the patient and his ventilator which are crucial for tolerance and acceptability. Recurrence plots provide a global view of these interactions and the Shannon entropy is shown to be a measure of the rate of asynchronisms as well as the breathing rhythm.