Sumiyoshi Abe

A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics

Abstract We show that a connection between the generalized entropy and theory of quantum groups, recently pointed out by Tsallis [Phys. Lett. A 195 (1994) 329], can naturally be understood in the framework of q -calculus. We present a new entropy which has q ↔ q −1 invariance and discuss its basic properties.

Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation

The special affine Fourier transformation (SAFT) is a generalization of the fractional Fourier transformation (FRT) and represents the most general lossless inhomogeneous linear mapping, in phase space, as the integral transformation of a wave function. Here we first summarize the most well-known optical operations on light-wave functions (i.e., the FRT, lens transformation, free-space propagation, and magnification), in a unified way, from the viewpoint of the one-parameter Abelian subgroups of the SAFT. Then we present a new operation, which is the Lorentz-type hyperbolic transformation in phase space and exhibits squeezing. We also show that the SAFT including these five operations can be generated from any two independent operations.

Axioms and uniqueness theorem for Tsallis entropy

Abstract The Shannon-Khinchin axioms for the ordinary information entropy are generalized in a natural way to the nonextensive systems based on the concept of nonextensive conditional entropy and a complete proof of the uniqueness theorem for the Tsallis entropy is presented. This improves the discussion of dos Santos [J. Math Phys. 38 (1997) 4104].

Stability of Tsallis entropy and instabilities of Rényi and normalized Tsallis entropies: A basis for q-exponential distributions

The q-exponential distributions, which are generalizations of the Zipf-Mandelbrot power-law distribution, are frequently encountered in complex systems at their stationary states. From the viewpoint of the principle of maximum entropy, they can apparently be derived from three different generalized entropies: the Rényi entropy, the Tsallis entropy, and the normalized Tsallis entropy. Accordingly, mere fittings of observed data by the q-exponential distributions do not lead to identification of the correct physical entropy. Here, stabilities of these entropies, i.e., their behaviors under arbitrary small deformation of a distribution, are examined. It is shown that, among the three, the Tsallis entropy is stable and can provide an entropic basis for the q-exponential distributions, whereas the others are unstable and cannot represent any experimentally observable quantities.

Scale-free statistics of time interval between successive earthquakes

The statistical property of the calm times, i.e., time intervals between successive earthquakes with arbitrary values of magnitude, is studied by analyzing the seismic time series data in California and Japan. It is found that the calm times obey the Zipf–Mandelbrot power law, exhibiting a new scale-free nature of the earthquake phenomenon. Dependence of the exponent of the power-law distribution on threshold for magnitude is examined. As threshold increases, the tail of the distribution tends to become longer, showing difficulty in statistically estimating time intervals of earthquakes.

Correlation induced by Tsallis’ nonextensivity

In Tsallis’ generalized statistical mechanics, correlation is induced by nonextensivity even if the microscopic degrees of freedom are dynamically independent. Here, using the classical ideal gas model, the generalized variance, covariance and correlation coefficient regarding the particle energies are calculated and their properties discussed. It is shown that the correlation is suppressed for a large number of particles. This demonstrates the validity of the independent particle picture for a dense gas rather than for a dilute gas. It is also found that, in the thermodynamic limit, the correlation again vanishes and the generalized variance exhibits a power-law behavior with respect to the particle number density. Relevance of these results to the zeroth law of thermodynamics in nonextensive statistical mechanics is pointed out.

Implications of Form Invariance to the Structure of Nonextensive Entropies

The form invariance of the statement of the maximum entropy principle and the metric structure in quantum density matrix theory, when generalized to nonextensive situations, is shown here to determine the structure of the nonextensive entropies. This limits the range of the nonextensivity parameter

General pseudoadditivity of composable entropy prescribed by the existence of equilibrium.

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Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach

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Anomalous diffusion in view of Einstein's 1905 theory of Brownian motion

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