5,578 research outputs found
Group entropies, correlation laws and zeta functions
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
Two-level electron-phonon systems with reflection symmetry linearly coupled
to one or two phonon modes (exciton and E 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
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
, 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
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
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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