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PubMed2005

Entropy factor in the hopping frequency for ionic conduction in oxide glasses induced by energetic clustering.

The contribution of configurational entropy to the effective hopping frequency of ionic transport in amorphous systems is discussed. The effective rate of ion hopping has been extracted from the onset frequency of the ac conductivity measured in ionically conducting silicate glasses. Both the onset frequency and the dc conductivity exhibit Arrhenius-type thermal activation with similar values for the activation energy, DeltaEa=0.65+/-0.3 eV. The prefactor of the onset frequency results in nu0'=(1.05+/-0.05)x10(11) Hz, which is much lower than characteristic vibrational frequencies (10(13) Hz). Following standard hopping percolation theory, the long-range motion is dominated by a fraction of high-energy barriers that connect clusters of faster sites. The multiplicity of equivalent sites for ion hop entails a retardation of the effective jumping time with respect to the elementary hop. This effect can be assimilated into a negative activation entropy term in the frequency prefactor of the ion hopping rate, which depends on the features of energy clustering and accounts for the wide dispersion of nu0' reported for many conducting glasses. The model implies an effective percolation length of Lc approximately 7 nm, in good agreement with previous works.

Journal Article↗

PubMed2005

Entropy- or enthalpy-driven collapse of strongly charged polymer chains in a one-component charged fluid of counterions or coions.

We applied a simulation method [T. Sumi and H. Sekino, J. Chem. Phys. 122, 194910 (2005)] to an infinitely dilute polyelectrolyte immersed in one-component charged fluids in order to investigate salt effects on its collapse. In this model system, the degree of freedom of the counterion (or the coion) is considered using a density-functional theory for polymer-solvent admixtures, while the oppositely charged ions are treated as a structureless background having the opposite charge. Results obtained by these simulations show that not only the counterion but also the coion makes the polymer chain collapsed. The effects by the coion are stronger than that by the counterion. Temperature variation of the gyration radius of the polymer chain immersed in the counterion is opposite to that in the coion: while the radius of gyration decreases as the temperature decreases in the case of the counterion, it decreases as the temperature increases in the case of the coion. From these results we conclude that the former is interpreted as an enthalpy-driven collapse caused by the screening effects of the counterion, whereas the latter is interpreted as an entropy-driven one due to the translational entropy of the coion.

Journal Article↗

PubMed2005

Information entropy, information distances, and complexity in atoms.

Shannon information entropies in position and momentum spaces and their sum are calculated as functions of Z(2 < or = Z < or = 54) in atoms. Roothaan-Hartree-Fock electron wave functions are used. The universal property S = a + b ln Z is verified. In addition, we calculate the Kullback-Leibler relative entropy, the Jensen-Shannon divergence, Onicescu's information energy, and a complexity measure recently proposed. Shell effects at closed-shell atoms are observed. The complexity measure shows local minima at the closed-shell atoms indicating that for the above atoms complexity decreases with respect to neighboring atoms. It is seen that complexity fluctuates around an average value, indicating that the atom cannot grow in complexity as Z increases. Onicescu's information energy is correlated with the ionization potential. Kullback distance and Jensen-Shannon distance are employed to compare Roothaan-Hartree-Fock density distributions with other densities of previous works.

Journal Article↗

PubMed2006

Entropy production during reversible polymerization in nonideal systems.

A general route is shown to calculate the entropy production sigma as function of time t in a closed system during reversible polymerization. We treat the polymer molecules to behave nonideal and apply exemplarily the classical Flory-Huggins theory to get explicit expressions for the activity coefficient. At the beginning of the polymerization the system is in a nonequilibrium state where chemical reactions take place that irreversibly drive the system towards equilibrium with sigma approaching zero in the limit t-->infinity. The time-dependent course of the entropy production is explicitly calculated for two cases where the reaction starts (i) from monomer molecules polymerizing to a defined number average chain length xn,eq and (ii) from monodisperse polymer molecules reacting with each other under the constrain that xn is the same at the beginning and the end of the reaction. In both cases we find that the nature of the activity coefficient has an important effect on the curvature of sigma which may considerably differ from that of an ideal behavior.

Journal Article↗

PubMed2006

On the analysis of Liquid Crystal NMR dipolar coupling data via mixed a priori-maximum entropy methods.

In this Letter, a general expression is derived for the conformational distribution function of a molecule dissolved in an anisotropic condensed fluid medium by combining an a priori model with the maximum entropy principle applied to treatment of liquid crystalline-NMR data. The recently proposed additive potential maximum entropy (APME) method is recovered as a special case, when the AP is chosen as the a priori model and the orientational order is low.

Letter↗

PubMed2006

Non-Gaussian energy landscape of a simple model for strong network-forming liquids: Accurate evaluation of the configurational entropy.

We present a numerical study of the statistical properties of the potential energy landscape of a simple model for strong network-forming liquids. The model is a system of spherical particles interacting through a square-well potential, with an additional constraint that limits the maximum number of bonds Nmax per particle. Extensive simulations have been carried out as a function of temperature, packing fraction, and Nmax. The dynamics of this model are characterized by Arrhenius temperature dependence of the transport coefficients and by nearly exponential relaxation of dynamic correlators, i.e., features defining strong glass-forming liquids. This model has two important features: (i) Landscape basins can be associated with bonding patterns. (ii) The configurational volume of the basin can be evaluated in a formally exact way, and numerically with an arbitrary precision. These features allow us to evaluate the number of different topologies the bonding pattern can adopt. We find that the number of fully bonded configurations, i.e., configurations in which all particles are bonded to Nmax neighbors, is extensive, suggesting that the configurational entropy of the low temperature fluid is finite. We also evaluate the energy dependence of the configurational entropy close to the fully bonded state and show that it follows a logarithmic functional form, different from the quadratic dependence characterizing fragile liquids. We suggest that the presence of a discrete energy scale, provided by the particle bonds, and the intrinsic degeneracy of fully bonded disordered networks differentiates strong from fragile behavior.

Journal Article↗

PubMed2006

Relative Boltzmann entropy, evolution equations for fluctuations of thermodynamic intensive variables, and a statistical mechanical representation of the zeroth law of thermodynamics.

Generalized thermodynamics or extended irreversible thermodynamics presumes the existence of thermodynamic intensive variables (e.g., temperature, pressure, chemical potentials, generalized potentials) even if the system is removed from equilibrium. It is necessary to properly understand the nature of such intensive variables and, in particular, of their fluctuations, that is, their deviations from those defined in the extended irreversible thermodynamic sense. The meaning of temperature is examined by means of a kinetic theory of macroscopic irreversible processes to assess the validity of the generalized (or extended) thermodynamic method applied to nonequilibrium phenomena. The Boltzmann equation is used for the purpose. Since the relative Boltzmann entropy has been known to be intimately related to the evolution of the aforementioned fluctuations in the intensive thermodynamic variables, we derive the evolution equations for such fluctuations of intensive variables to lay the foundation for investigating the physical implications and evolution of the relative Boltzmann entropy, so that the range of validity of the thermodynamic theory of irreversible processes can be elucidated. Within the framework of this work, we examine a special case of the evolution equations for the aforementioned fluctuations of intensive variables, which also facilitate investigation of the molecular theory meaning of the zeroth law of thermodynamics. We derive an evolution equation describing the relaxation of temperature fluctuations from its local value and present a formula for the temperature relaxation time.

Journal Article↗

PubMed2006

A mechanical representation of entropy for a large finite system.

The temporal evolution of the entropy of a mechanical system as described by a single trajectory is computed using the Clausius [Philos. Mag. 40, 122 (1868)] equality. This requires computing the maximal work that can be done by the system and comparing it to the actual work performed. A single trajectory suffices to determine the entropy when it is "typical," meaning that average values of mechanical variables will not be different when computed using trajectories with different initial conditions. The results are illustrated for small rare gas clusters heated and compressed by an impact at a hard surface.

Journal Article↗

PubMed1969

Geometric programming, chemical equilibrium, and the anti-entropy function.

THE CULMINATION OF THIS PAPER IS THE FOLLOWING DUALITY PRINCIPLE OF THERMODYNAMICS: maximum S = minimum S(*). (1) The left side of relation (1) is the classical characterization of equilibrium. It says to maximize the entropy function S with respect to extensive variables which are subject to certain constraints. The right side of (1) is a new characterization of equilibrium and concerns minimization of an anti-entropy function S(*) with respect to intensive variables. Relation (1) is applied to the chemical equilibrium of a mixture of gases at constant temperature and volume. Then (1) specializes to minimum F = maximum F(*), (2) where F is the Helmholtz function for free energy and F(*) is an anti-Helmholtz function. The right-side of (2) is an unconstrained maximization problem and gives a simplified practical procedure for calculating equilibrium concentrations. We also give a direct proof of (2) by the duality theorem of geometric programming. The duality theorem of geometric programming states that minimum cost = maximum anti-cost. (30).

Journal Article↗

PubMed1974

Entropy, dynamics, and molecular chaos.

With the help of simple probabilistic models of Kac and McKean, we discuss the meaning of the generalized expression for entropy that was recently introduced by our group and compare it with Boltzmann's expression. We emphasize the fact that Boltzmann's formulation in terms of the single particle distribution function, f(1), requires very restricted assumptions about the preparation of the system (chaos) and the nature of the collision mechanism (Markov processes).Our generalized [unk]-theorem, however, refers to the complete system; in general, it does not lead to an [unk]-theorem for the single particle distribution function, f(1). It is valid whatever the preparation of the system. In McKean's model, situations exist where it gives the correct behavior while the Boltzmann's expression for entropy becomes meaningless. In addition, in Kac's model, we show that correlations reach equilibrium more rapidly than f(1) and that there is an asymptotic regime where both formulations give the same result.

Journal Article↗

PubMed1975

Partial knowledge, entropy, and estimation.

In a growing body of literature, available partial knowledge is used to estimate the prior probability distribution p identical with(p(1),...,p(n)) by maximizing entropy H(p) identical with-Sigmap(i) log p(i), subject to constraints on p which express that partial knowledge. The method has been applied to distributions of income, of traffic, of stock-price changes, and of types of brand-article purchases. We shall respond to two justifications given for the method:(alpha) It is "conservative," and therefore good, to maximize "uncertainty," as (uniquely) represented by the entropy parameter.(beta) One should apply the mathematics of statistical thermodynamics, which implies that the most probable distribution has highest entropy.Reason (alpha) is rejected. Reason (beta) is valid when "complete ignorance" is defined in a particular way and both the constraint and the estimator's loss function are of certain kinds.

Journal Article↗

PubMed1978

Nonequilibrium entropy, Lyapounov variables, and ergodic properties of classical systems.

We discuss the problem of defining (nonequilibrium) entropy in terms of the concepts of mechanics and of reconciling its monotonic increase with the Hamiltonian evolution of the dynamical system. This leads to investigating necessary and sufficient conditions for the existence of monotonically increasing quantities or the so-called Lyapounov variables of classical systems. It is found that the condition of "mixing" is necessary and the property of being K-flow is sufficient for the existence of a Lyapounov variable. The significance of the study of Lyapounov variables for the elucidation of the fundamental questions of statistical mechanics is briefly discussed. It is seen that every Lyapounov variable must fail to commute with at least some of the operators of multiplication by phase space functions. The uncertainty relations implied by this necessary noncommutativity would then set a limit on the simultaneous determination of entropy and trajectories in phase space. These considerations thus support and sharpen the view that the thermodynamical and the (microscopic) dynamical descriptions of classical systems could be consistently reconciled as being complementary descriptions analogous to the complementary descriptions encountered in quantum mechanics.

Journal Article↗

PubMed1981

On the entropy function in sociotechnical systems.

The entropy function H = -Sigmap(j) log p(j) (p(j) being the probability of a system being in state j) and its continuum analogue H = integralp(x) log p(x) dx are fundamental in Shannon's theory of information transfer in communication systems. It is here shown that the discrete form of H also appears naturally in single-lane traffic flow theory. In merchandising, goods flow from a whole-saler through a retailer to a customer. Certain features of the process may be deduced from price distribution functions derived from Sears Roebuck and Company catalogues. It is found that the dispersion in logarithm of catalogue prices of a given year has remained about constant, independently of the year, for over 75 years. From this it may be inferred that the continuum entropy function for the variable logarithm of price had inadvertently, through Sears Roebuck policies, been maximized for that firm subject to the observed dispersion.

Journal Article↗

PubMed1990

Application of entropy measures derived from the ergodic theory of dynamical systems to rat locomotor behavior.

Measures of complexity derived from ergodic theory of dynamical systems were developed and applied to an exemplary data set describing locomotor movements of rats in a bounded space. A symbolic dynamical system was obtained by partitioning the event space into equally probable partition elements, using a k-dimensional tree. The measures calculated from the symbolic sequences included the topological entropy (ht)--i.e., the rate of increase of all possible sequences with increasing sequence length--and the metric entropy (hm)--i.e., the rate of increase of all likely sequences with increasing sequence length. These measures were used to assess changes in rat locomotor behavior as recorded in the behavioral pattern monitor (BPM) that were induced by amphetamine (0.25, 0.50, 1.0, or 2.0 mg/kg) and 3,4-methylenedioxymethamphetamine (MDMA; 1.25, 2.5, 5.0, or 10.0 mg/kg). Amphetamine increased the mean activity, ht, and hm. MDMA resulted in a monotonic dose-response curve for activity but exhibited a biphasic dose response in ht and hm. In particular, some animals in the higher dose groups showed a ht in the range of the saline controls, whereas other animals exhibited a significantly reduced ht and a greater decrease in hm, suggesting that two different behavioral reactions coexist within the same higher dose range of MDMA. An important implication of our method is that, in applied ergodic measure-theoretic approaches, the partition that determines the elements of the symbolic dynamical system should not be specified a priori on abstract mathematical grounds but should be chosen relative to its significance with respect to the data set in question. Here, the animal constructs its own spatiotemporal partition in behavioral phase space.

3,4-Methylenedioxyamphetamine↗

PubMed1992

On the origin of the enthalpy and entropy convergence temperatures in protein folding.

Temperature dependence of the thermodynamics of folding/unfolding for cytochrome c has been determined as a function of moderate [0-10% (vol/vol)] concentrations of methanol. Heat capacity change (delta Cp) for unfolding decreases with increased concentrations of methanol, consistent with a higher solvent hydrophobicity. For a given transition temperature, this effect results in higher experimental enthalpy (delta H) and entropy (delta S) changes with increased methanol concentrations. When the enthalpy or entropy data sets obtained at different methanol concentrations are plotted as a function of temperature, they are seen to converge and assume common values around 100 degrees C for delta H and 112 degrees C for delta S. These convergence temperatures are similar to those obtained for different proteins in aqueous solution when delta H and delta S are normalized with respect to number of residues. It has been previously hypothesized that these convergence temperatures correspond to the temperatures at which the hydrophobic contributions to delta H and delta S are zero; the results presented here agree with this viewpoint.

Calorimetry, Differential Scanning↗

PubMedDate not supplied

The use of regularity as estimated by approximate entropy to distinguish saltatory growth.

A nonlinear dynamics metric, approximate entropy (ApEn), is investigated as a diagnostic method for distinguishing between mathematical models, and the underlying mechanistic hypotheses that purport to describe the same time series experimental observations. ApEn measures the occurrence of pattern regularity within a time series, and is used here to investigate growth patterns in daily length growth. The notion investigated is that ApEn distributions for competing time series patterns expressed as mathematical formulations can be modelled by Monte Carlo and bootstrap methods and compared to the ApEn values for an original experimental data series. If the ApEn values for the different models do not overlap, then it is expected that ApEn can be utilized to distinguish these models and hypotheses, and to provide statistical assessment for the underlying biological patterns in experimental data. The conclusion is that the ApEn metric is successful as a time series diagnostic tool. It is a model-independent statistic that clearly differentiates saltatory growth from slowly varying continuous models of growth and serves to further document the saltatory nature of growth. This is a unique application of approximate entropy, illustrating the broad applicability of ApEn to biological time series, with the specific example of discriminating a saltatory growth process in longitudinal growth data. Future investigations of regularity in longitudinal time series in human biology with ApEn statistics are suggested.

Data Interpretation, Statistical↗

PubMed1995

Deconvolution of planar scintigrams by maximum entropy.

Planar scintigrams are deconvolved with a point spread function using the maximum entropy method with the aim of improving image quality. The technique requires the specification of several parameters. These are related to the level of noise present in the data and our a priori knowledge of the object imaged. The performance of the technique is tested for a wide range of these parameters using images of a Williams phantom in scattering material and a figure of merit, derived from the detectability of the smallest cold spot, is calculated. For close to optimal values of the parameters a factor of two improvement in the figure is found. A processed bone image shows improved contrast and resolution. Maximum entropy processing could be used to increase image quality or allow comparable image quality with reduced imaging time or patient dose.

Bone and Bones↗

PubMed1996

Development of low entropy coding in a recurrent network.

In this paper we present an unsupervised neural network which exhibits competition between units via inhibitory feedback. The operation is such as to minimize reconstruction error, both for individual patterns, and over the entire training set. A key difference from networks which perform principal components analysis, or one of its variants, is the ability to converge to non-orthogonal weight values. We discuss the network's operation in relation to the twin goals of maximizing information transfer and minimizing code entropy, and show how the assignment of prior probabilities to network outputs can help to reduce entropy. We present results from two binary coding problems, and from experiments with image coding.

Journal Article↗