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At least 1,117 records · Page 62Linked to original sources
PubMed2002

Morphology from the maximum entropy principle: domains in a phase ordering system and a crack pattern in broken glass.

The maximum entropy principle is applied to study the morphology of a phase ordering two-dimensional system below the critical point. The distribution of domain area A is a function of ratio of the area to contour length L, R=A/L(A), and is given by exp(-lambda R(mu)) with exponent mu=2, which follows from the Lifshitz-Cahn-Allen theory. A and L are linked through the relation L approximately A(nu). We find two types of domain in the system: large of elongated shape (nu=0.88) and small of circular shape (nu=0.5). A crack pattern in broken glass belongs to the same morphology class with mu=1 and nu=0.72.

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PubMed2002

Recursive approach for constructing the q=1/2 maximum entropy distribution from redundant data.

A recursive approach for computing the q=1/2 nonextensive maximum entropy distribution of the previously introduced formalism for data subset selection is proposed. Such an approach is based on an iterative biorthogonalization technique, which allows for the incorporation of the Lagrange multipliers that determine the distribution to the workings of the algorithm devised for selecting relevant data subsets. This technique circumvents the necessity of inverting operators and yields a recursive procedure to appropriately modify the Lagrange multipliers so as to account for each new constraint.

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PubMed2002

Temperature and pressure dependence of the alpha relaxation and configurational entropy of a prototype glass former.

The alpha relaxation process of the fragile glass former salol is investigated in the T-P domain by means of photon correlation spectroscopy. We find that a time-pressure superposition principle is obeyed for the relaxation function in addition to the time-temperature superposition. The behavior of the relaxation time is studied by using an extended version of the Adam-Gibbs model including the pressure dependence. The excellent conformity of the equation describing the bidimensional surface tau(T,P) to the experimental data provides a positive check for this model, here verified on photon correlation measurements. The same model gives a rationale of the phenomenological expressions recently introduced to describe the changes in the slow dynamics induced by varying both temperature and pressure. These findings suggest that the reduction of configurational entropy actually guides the liquid toward the glass transition.

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PubMed2003

Noise-level estimation of time series using coarse-grained entropy.

We present a method of noise-level estimation that is valid even for high noise levels. The method makes use of the functional dependence of coarse-grained correlation entropy K2(epsilon ) on the threshold parameter epsilon. We show that the function K2(epsilon ) depends, in a characteristic way, on the noise standard deviation sigma. It follows that observing K2 (epsilon ) one can estimate the noise level sigma. Although the theory has been developed for the Gaussian noise added to the observed variable we have checked numerically that the method is also valid for the uniform noise distribution and for the case of Langevin equation corresponding to the dynamical noise. We have verified the validity of our method by applying it to estimate the noise level in several chaotic systems and in the Chua electronic circuit contaminated by noise.

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PubMed2003

Stability analysis of mean-field-type nonlinear Fokker-Planck equations associated with a generalized entropy and its application to the self-gravitating system.

Multidimensional nonlinear Fokker-Planck equations of mean-field type are proposed within the framework of generalized thermostatistics to develop a general formulation of stability analysis of their solutions. Two types of eigenvalue equations are studied. The nonlinear Fokker-Planck equations are shown to exhibit an H theorem with a Liapunov functional that takes the form of a free energy involving generalized entropies of Tsallis. The second-order variation of the Liapunov functional is computed to conduct local stability analysis and the associated eigenvalue equation is derived for an arbitrary form of mean-field coupling potential. Assuming quasiequilibrium for the velocity distribution, the reduced eigenvalue equation with space coordinates alone is also obtained. The alternative type of eigenvalue equation based on the linearization of the nonlinear Fokker-Planck equations is presented. Taking the mean-field coupling potential to be the gravitational one, the nonlinear Fokker-Planck equation in terms of three-dimensional velocity and space coordinates together with the framework of stability analysis is shown to be applicable to a mean-field model of self-gravitating system. By solving the eigenvalue equation for the eigenfunction with 0 eigenvalue, the occurrence of stability change of the equilibrium probability density with spherical symmetry is discussed.

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PubMed2003

Fractal entropy of a chain of nonlinear oscillators.

We study the time evolution of a chain of nonlinear oscillators. We focus on the fractal features of the spectral entropy and analyze its characteristic intermediate time scales as a function of the nonlinear coupling. A Brownian motion is recognized with an analytic power-law dependence of its diffusion coefficient on the coupling.

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PubMed2003

Non-Gaussian probability distribution functions from maximum-entropy-principle considerations.

In this work we develop the recently suggested concept of superstatistics [C. Beck and E.G.D. Cohen, Physica A 322, 267 (2003)], face the problem of devising a viable way for estimating the correct statistics for a system in the absence of sufficient knowledge of its microscopical dynamics, and suggest to solve it through the maximum-entropy principle. As an example, we deduce the probability distribution function for velocity fluctuations in turbulent fluids, which is slightly different from the form suggested by C. Beck [Phys. Rev. Lett. 87, 180601 (2001)].

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PubMed2003

Maximum entropy theory and the rapid relaxation of three-dimensional quasi-geostrophic turbulence.

Turbulent flow in a rapidly rotating stably stratified fluid (quasi-geostrophic turbulence) commonly decays toward a stable pattern of large-scale jets or vortices. A formula for the most probable three-dimensional end state, the maximum entropy state (MES), is derived using a form of Lynden-Bell statistical mechanics. The MES is determined by a set of integral invariants, including energy, as opposed to a complete description of the initial condition. A computed MES qualitatively resembles the quasistationary end state of a numerical simulation that is initialized with red noise, and relaxes for a time on the order of 100 (initial) eddy turnovers. However, the potential enstrophy of the end state, obtained from a coarsened potential vorticity distribution, exceeds that of the MES by nearly a factor of 2. The estimated errors for both theory and simulation do not account for the discrepancy. This suggests that the MES, if ever realized, requires a much longer time scale to fully develop.

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PubMed2004

Comment on "Critique of q-entropy for thermal statistics".

It was recently argued [M. Nauenberg, Phys. Rev. E 67, 036114 (2003)] that the theory sometimes referred to as nonextensive statistical mechanics has no physical basis, for a considerable variety of reasons, including the impossibility of measuring the temperature out of the Boltzmann-Gibbs (BG) theory. We comment here on virtually all the physically and mathematically relevant issues, and point out what we consider to be severe inadvertences contained in that paper. In particular, we factually argue, through computer simulations, the validity of the zeroth principle of thermodynamics, and of the basic rules of thermometry for nonextensive systems. This fact further supports the possible connection with the thermodynamics of nonextensive statistical mechanics, which is already known to be consistent with the first, second, and third principles. All the foundational steps (e.g., the uniqueness of the entropy and the stationary state distribution) have already been established for nonextensive thermostatistics on similar grounds than those long known for BG statistics, the former corresponding to power laws (expected for long-range interactions when size N diverges before time t), and the latter correspond to the BG exponential law (expected for long-range interactions when N diverges after t, as well as for short-range interactions in any diverging order for N and t). We conclude that the invalidating arguments made by Nauenberg by no means apply.

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PubMed2004

Comment on "Mobility spectrum computational analysis using a maximum entropy approach".

We point out that the comparison in Fig. 1 of the recent publication by S. Kiatgamolchai et al. [Phys. Rev. E 66, 036705 (2002)] of the proposed maximum entropy-mobility spectrum analysis (ME-MSA) with our quantitative mobility spectrum analysis (QMSA) is misleading. Rather than comparing with the more recent "improved" version of QMSA [Vurgaftman et al., J. Appl. Phys. 84, 4966 (1998)], a preliminary version that was three years older and demonstrably inferior was employed. We show that ME-MSA and the improved QMSA give quite similar results.

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PubMed2004

Rényi entropies and observables.

Evidence is given that Rényi entropies of macroscopic thermodynamic systems defined on the bases of probabilities of microstates cannot be related to observables. The notion of observable is clarified.

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PubMed2004

Constructive approximations to the q=1/2 maximum entropy distribution from redundant and noisy data.

The problem of constructing the q=1/2 nonextensive maximum entropy distributions from redundant and noisy data is considered. A strategy is proposed, which evolves through the following steps. (i) Independent constraints are first preselected by recourse to a data-independent technique to be discussed here. (ii) The data are a posteriori used to determine the parameters of the distribution by a previously introduced forward approach. (iii) A backward approach is proposed for reducing the parameters of such distribution. The previously introduced forward approach is generalized here in order to make it suitable for dealing with very noisy data.

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PubMed2004

Detecting dynamical changes in time series using the permutation entropy.

Timely detection of unusual and/or unexpected events in natural and man-made systems has deep scientific and practical relevance. We show that the recently proposed conceptually simple and easily calculated measure of permutation entropy can be effectively used to detect qualitative and quantitative dynamical changes. We illustrate our results on two model systems as well as on clinically characterized brain wave data from epileptic patients.

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PubMed2004

Granular contact force density of states and entropy in a modified Edwards ensemble.

A method has been found to analyze Edwards' granular contact force probability functional for a special case. As a result, the granular contact force probability density functions are obtained from first principles for this case. The results are in excellent agreement with the experimental and simulation data. The derivation assumes Edwards' flat measure--a density of states (DOS) that is uniform within the metastable regions of phase space. The enabling assumption, supported by physical arguments and empirical evidence, is that correlating information is not significantly recursive through loops in the packing. Maximizing a state-counting entropy results in a transport equation that can be solved numerically. For the present this has been done using the "mean-structure approximation," projecting the DOS across all angular coordinates to more clearly identify its predominant nonuniformities. These features are (1) the grain factor Psi related to grain stability and strong correlation between the contact forces on the same grain and (2) the structure factor Upsilon related to Newton's third law and strong correlation between neighboring grains.

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PubMed2005

Some properties of the entropy in the natural time.

We show that the entropy S , defined as S identical with chi ln chi - chi ln chi [Phys. Rev. E 68, 031106 (2003)] where chi stands for the natural time [Phys. Rev. E 66, 011902 (2002)], exhibits positivity and concavity as well as stability or experimental robustness. Furthermore, the distinction between the seismic electric signal activities and "artificial" noises, based on the classification of their S values, is lost when studying the time-reversed signals. This reveals the profound importance of considering the (true) time arrow.

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PubMed2005

Fluctuation-dissipation relation and the Edwards entropy for a glassy granular compaction model.

We analytically study a one-dimensional compaction model in the glassy regime. Both correlation and response functions are calculated exactly in the evolving dense and low tapping strength limit, where the density relaxes in a 1/ln t fashion. The response and correlation functions turn out to be connected through a nonequilibrium generalization of the fluctuation-dissipation theorem. The initial response in the average density to an increase in the tapping strength is shown to be negative, while on longer time scales it is shown to be positive. On short time scales the fluctuation-dissipation theorem governs the relation between correlation and response, and we show that such a relationship also exists for the slow degrees of freedom, albeit with a different temperature. The model is further studied within the statistical theory proposed by Edwards and co-workers, and the Edwards entropy is calculated in the large system limit. The fluctuations described by this approach turn out to match the fluctuations as calculated through the dynamical consideration. We thus have an instance where these ideas can be confirmed analytically in a non-mean-field model.

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PubMed2006

Continuous phase transitions with a convex dip in the microcanonical entropy.

The appearance of a convex dip in the microcanonical entropy of finite systems usually signals a first order transition. However, a convex dip also shows up in some systems with a continuous transition as, for example, in the Baxter-Wu model and in the four-state Potts model in two dimensions. We demonstrate that the appearance of a convex dip in those cases can be traced back to a finite-size effect. The properties of the dip are markedly different from those associated with a first order transition and can be understood within a microcanonical finite-size scaling theory for continuous phase transitions. Results obtained from numerical simulations corroborate the predictions of the scaling theory.

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