5,578 research outputs found

    Group entropies, correlation laws and zeta functions

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    The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.Comment: to appear in Physical Review

    Entropic uncertainty measure for fluctuations in two-level electron-phonon models

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    Two-level electron-phonon systems with reflection symmetry linearly coupled to one or two phonon modes (exciton and E⊗(b1+b2)\otimes(b_1+b_2) Jahn-Teller model) exhibit strong enhancement of quantum fluctuations of the phonon coordinates and momenta due to the complex interplay of quantum fluctuations and nonlinearities inherent to the models. We show that for the complex correlated quantum fluctuations of the anisotropic two-level systems the Shannon entropies of phonon coordinate and momentum and their sum yield their proper global description. On the other hand, the variance measures of the Heisenberg uncertainties suffer from several shortcomings to provide proper description of the fluctuations. Wave functions, related entropies and variances were determined by direct numerical simulations. Illustrative variational calculations were performed to demonstrate the effect on an analytically tractable exciton model.Comment: 14 pages, 10 figs, published in Eur.Phys.J 38 B (2004) 25-3

    Generalised asymptotic equivalence for extensive and non-extensive entropies

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    We extend the Hanel and Thurner asymptotic analysis to both extensive and non-extensive entropies on the basis of a wide class of entropic forms. The procedure is known to be capable to classify multiple entropy measures in terms of their defining equivalence classes. Those are determined by a pair of scaling exponents taking into account a large number of microstates as for the thermodynamical limit. Yet, a generalisation to this formulation makes it possible to establish an entropic connection between Markovian and non-Markovian statistical systems through a set of fundamental entropies S±S_{\pm}, which have been studied in other contexts and exhibit, among their attributes, two interesting aspects: They behave as additive for a large number of degrees of freedom while they are substantially non-additive for a small number of them. Furthermore, an ample amount of special entropy measures, either additive or non-additive, are contained in such asymptotic classification. Under this scheme we analyse the equivalence classes of Tsallis, Sharma-Mittal and R\'enyi entropies and study their features in the thermodynamic limit as well as the correspondences among them.Comment: 6 pages, 2 figure

    Noise and disturbance in quantum measurements: an information-theoretic approach

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    We introduce information-theoretic definitions for noise and disturbance in quantum measurements and prove a state-independent noise-disturbance tradeoff relation that these quantities have to satisfy in any conceivable setup. Contrary to previous approaches, the information-theoretic quantities we define are invariant under relabelling of outcomes, and allow for the possibility of using quantum or classical operations to `correct' for the disturbance. We also show how our bound implies strong tradeoff relations for mean square deviations.Comment: v3: to appear on PRL (some issues fixed, supplemental material expanded). v2: replaced with submitted version; 5 two-column pages + 6 one-column pages + 3 figures; one issue corrected and few references added. v1: 17 pages, 3 figure

    Shannon entropies of atomic structure factors, off-diagonal order and electron correlation

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    Shannon entropies of one- and two-electron atomic structure factors in the position and momentum representations are used to examine the behavior of the off-diagonal elements of density matrices with respect to the uncertainty principle and to analyze the effects of electron correlation on off-diagonal order. We show that electron correlation induces off-diagonal order in position space which is characterized by larger entropic values. Electron correlation in momentum space is characterized by smaller entropic values as information is forced into regions closer to the diagonal. Related off-diagonal correlation functions are also discussed
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