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Sep 18, 2013
09/13

by
Sumiyoshi Abe

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Macroscopic nonextensive thermodynamics is studied without recourse to microscopic statistical mechanics. It is shown that if entropy is nonextensive, the concept of physical temperature introduced through the generalized zeroth law of thermodynamics necessarily leads to modifications of the first law of thermodynamics and some of thermodynamic relations including Clausius definition of thermodynamic entropy. It is also shown, by applying this generalized Clausius entropy to a composite...

Source: http://arxiv.org/abs/cond-mat/0012115v3

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Sep 17, 2013
09/13

by
Sumiyoshi Abe

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It is shown how, among a class of generalized entropies, the Tsallis entropy can uniquely be identified by the principles of thermodynamics, the concept of stability and the axiomatic foundation.

Source: http://arxiv.org/abs/cond-mat/0305087v1

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Sep 19, 2013
09/13

by
Sumiyoshi Abe

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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 Renyi 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...

Source: http://arxiv.org/abs/cond-mat/0206078v3

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Sep 20, 2013
09/13

by
Sumiyoshi Abe

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The Kullback-Leibler divergence offers an information-theoretic basis for measuring the difference between two given distributions. Its quantum analog, however, fails to play a corresponding role for comparing two density matrices, if the reference states are pure states. Here, it is shown that nonadditive (nonextensive) generalization of quantum information theory is free from such a difficulty and the associated quantity, termed the quantum q-divergence, can in fact be a good...

Source: http://arxiv.org/abs/quant-ph/0301136v1

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Sep 18, 2013
09/13

by
Sumiyoshi Abe

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q-Expectation value of a physical quantity is widely used in nonextensive statistical mechanics. Here, it is shown that the q-expectation value is not stable under small deformations of a probability distribution function, in general, whereas the ordinary expectation value is always stable.

Source: http://arxiv.org/abs/0806.3934v1

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Sep 17, 2013
09/13

by
Sumiyoshi Abe

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Given an original distribution, its statistical and probabilistic attributs may be scanned by the associated escort distribution introduced by Beck and Schlogl and employed in the formulation of nonextensive statistical mechanics. Here, the geometric structure of the one-parameter family of the escort distributions is studied based on the Kullback-Leibler divergence and the relevant Fisher metric. It is shown that the Fisher metric is given in terms of the generalized bit-variance, which...

Source: http://arxiv.org/abs/cond-mat/0305231v1

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Sep 20, 2013
09/13

by
Sumiyoshi Abe

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Nonadditive (nonextensive) generalization of the quantum Kullback-Leibler divergence, termed the quantum q-divergence, is shown not to increase by projective measurements in an elementary manner.

Source: http://arxiv.org/abs/quant-ph/0301137v3

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Sep 18, 2013
09/13

by
Sumiyoshi Abe

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It is pointed out that the Tsallis entropy functional represented in terms of the escort distribution is not concave of the entropic index $q$ is less than unity. It is emphasized that the escort distribution is a secondary object calculated from the basic original distribution.

Source: http://arxiv.org/abs/cond-mat/0006053v1

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Sep 22, 2013
09/13

by
Sumiyoshi Abe

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Based on the canonical formalism, the dilatation symmetry is implemented to the Fokker-Planck equation for the Wigner distribution function that describes atomic motion in an optical lattice. This reveals the symmetry principle underlying the recent result obtained by Lutz [Phys. Rev. A 67, 051402(R) (2003)] on the connection between anomalous transport in the optical lattice and Tsallis statistics in the high-energy regime.Lutz's discussion is generalized to the nonstationary case, and the...

Source: http://arxiv.org/abs/cond-mat/0307306v1

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Sep 18, 2013
09/13

by
Sumiyoshi Abe

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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.

Source: http://arxiv.org/abs/cond-mat/0005538v1

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Sep 18, 2013
09/13

by
Sumiyoshi Abe

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The concept of composability states that entropy of the total system composed of independent subsystems is a function of entropies of the subsystems. Here, the most general pseudoadditivity rule for composable entropy is derived based only on existence of equilibrium.

Source: http://arxiv.org/abs/cond-mat/0012206v1

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Sep 21, 2013
09/13

by
Sumiyoshi Abe

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Through the generalization of Khinchin's classical axiomatic foundation, a basis is developed for nonadditive information theory. The classical nonadditive conditional entropy indexed by the positive parameter q is introduced and then translated into quantum information. This quantity is nonnegative for classically correlated states but can take negative values for entangled mixed states. This property is used to study quantum entanglement in the parametrized Werner-Popescu-like state of an...

Source: http://arxiv.org/abs/quant-ph/0104135v1

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Sep 18, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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The district of southern California and Japan are divided into small cubic cells, each of which is regarded as a vertex of a graph if earthquakes occur therein. Two successive earthquakes define an edge and a loop, which replace the complex fault-fault interaction. In this way, the seismic data are mapped to a random graph. It is discovered that an evolving random graph associated with earthquakes behaves as a scale-free network of the Barabasi-Albert type. The distributions of connectivities...

Source: http://arxiv.org/abs/cond-mat/0210289v2

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Sep 23, 2013
09/13

by
Sumiyoshi Abe; Stefan Thurner

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Einstein's theory of Brownian motion is revisited in order to formulate generalized kinetic theory of anomalous diffusion. It is shown that if the assumptions of analyticity and the existence of the second moment of the displacement distribution are relaxed, the fractional derivative naturally appears in the diffusion equation. This is the first demonstration of the physical origin of the fractional derivative, in marked contrast to the usual phenomenological introduction of it. Furthermore,...

Source: http://arxiv.org/abs/cond-mat/0411645v1

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Sep 17, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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A part of the seismic time series, in which the Omori law for temporal pattern of aftershocks holds, is refereed to as the Omori regime. Here the properties of correlation between earthquake events both inside and outside of the Omori regime are studied by analysis of the data taken in southern California. It is found that inside of the Omori regime correlation exhibits the aging phenomenon, in marked contrast to the fact that no aging is observed outside of the Omori regime. The scaling nature...

Source: http://arxiv.org/abs/cond-mat/0305509v1

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Sep 22, 2013
09/13

by
Sumiyoshi Abe; Tsunehiro Kobayashi

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Microcanonical ensemble theory of bosons is derived from quantum mechanics by making use of a hidden gauge structure. The relative phase interaction associated with this gauge structure, described by the Pegg-Barnett formalism, is shown to lead to perfect decoherence in the thermodynamics limit and the principle of equal a priori probability, simultaneously.

Source: http://arxiv.org/abs/cond-mat/0205666v1

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Sep 20, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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The network approach plays a distinguished role in contemporary science of complex systems/phenomena. Such an approach has been introduced into seismology in a recent work [S. Abe and N. Suzuki, Europhys. Lett. 65, 581 (2004)]. Here, we discuss the dynamical property of the earthquake network constructed in California and report the discovery that the values of the clustering coefficient remain stationary before main shocks, suddenly jump up at the main shocks, and then slowly decay following a...

Source: http://arxiv.org/abs/physics/0612058v1

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Sep 17, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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The statistical properties of time intervals between significant earthquakes are found to be described by the Zipf-Mandelbrot-Tsallis-type distribution.

Source: http://arxiv.org/abs/cond-mat/0207657v1

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Sep 19, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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The Internet is a complex system, whose temporal behavior is highly nonstationary and exhibits sudden drastic changes regarded as main shocks or catastrophes. Here, analyzing a set of time series data of round-trip tim measured in echo experiment with the Ping Command, the property of "aftershocks" (i.e., catastrophes of smaller scales) after a main shock is studied. It is found that the aftershocks obey Omori's law. Thus, the Internet shares with earthquakes and financial market...

Source: http://arxiv.org/abs/cond-mat/0206453v1

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Sep 22, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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Discoveries of the scale-free and small-world features are reported on a network constructed from the seismic data. It is shown that the connectivity distribution decays as a power law, and the value of the degrees of separation, i.e., the characteristic path length or the diameter, between two earthquakes (as the vertices) chosen at random takes a small value between 2 and 3. The clustering coefficient is also calculated and is found to be about 10 times larger than that in the case of the...

Source: http://arxiv.org/abs/cond-mat/0308208v1

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Sep 20, 2013
09/13

by
Ugur Tirnakli; Sumiyoshi Abe

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Event correlation between aftershocks in the coherent noise model is studied by making use of natural time, which has recently been introduced in complex time-series analysis. It is found that the aging phenomenon and the associated scaling property discovered in the observed seismic data are well reproduced by the model. It is also found that the scaling function is given by the $q$-exponential function appearing in nonextensive statistical mechanics, showing power-law decay of event...

Source: http://arxiv.org/abs/cond-mat/0405398v1

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Sep 21, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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To characterize the dynamical features of seismicity as a complex phenomenon, the seismic data is mapped to a growing random graph, which is a small-world scale-free network. Here, hierarchical and mixing properties of such a network are studied. The clustering coefficient is found to exhibit asymptotic power-law decay with respect to connectivity, showing hierarchical organization. This structure is supported by not only main shocks but also small shocks, and may have its origin in the...

Source: http://arxiv.org/abs/cond-mat/0602076v1

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Sep 22, 2013
09/13

by
Sumiyoshi Abe; Stefan Thurner

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In analogy to superstatistics, which connects Boltzmann-Gibbs statistical mechanics to its generalizations through temperature fluctuations, complex networks are constructed from the fluctuating Erdos-Renyi random graphs. Here, using the quantum mechanical method, the exact analytic formula is presented for the hidden variable distribution, which describes the fluctuation and generates a generic degree distribution through the Poisson transformation. As an example, a static scale-free network...

Source: http://arxiv.org/abs/cond-mat/0501429v1

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Sep 20, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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The cumulative probability distribution of sparseness time interval in the Internet is studied by the method of data analysis. Round-trip time between a local host and a destination host through ten odd routers is measured using the Ping Command, i.e., doing echo experiment. It is found that the data are well described by the q-exponential destributions, which maximize the Tsallis entropy indexed by q less or larger than unity. The network is observed to itinerate over a series of the...

Source: http://arxiv.org/abs/cond-mat/0204336v2

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Sep 18, 2013
09/13

by
Sumiyoshi Abe; Satoshi Kaneko

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The concept of work is studied in quantum thermostatistics of a system surrounded by an environment and driven by an external force. It is found that there emerges the gauge theoretical structure in a nonequilibrium process, the field of which is referred to as the work gauge field. The thermodynamic work as the flux of the work gauge field is considered for a cyclic process in the space of the external-force parameters. As an example, the system of a spin-1/2 interacting with an external...

Source: http://arxiv.org/abs/cond-mat/0509603v1

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Sep 20, 2013
09/13

by
Sumiyoshi Abe; Stefan Thurner

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The Erdos-Renyi classical random graph is characterized by a fixed linking probability for all pairs of vertices. Here, this concept is generalized by drawing the linking probability from a certain distribution. Such a procedure is found to lead to a static complex network with an arbitrary connectivity distribution. In particular, a scale-free network with the hierarchical organization is constructed without assuming any knowledge about the global linking structure, in contrast to the...

Source: http://arxiv.org/abs/cond-mat/0601159v1

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Sep 18, 2013
09/13

by
Sumiyoshi Abe; J. Zak

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Based on the recently introduced averaging procedure in phase space, a new type of entropy is defined on the von Neumann lattice. This quantity can be interpreted as a measure of uncertainty associated with simultaneous measurement of the position and momentum observables in the discrete subset of the phase space. Evaluating for a class of the coherent states, it is shown that this entropy takes a stationary value for the ground state, modulo a unit cell of the lattice in such a class. This...

Source: http://arxiv.org/abs/quant-ph/0202092v1

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Sep 18, 2013
09/13

by
Yuichi Itto; Sumiyoshi Abe

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Wegner's method of flow equations offers a useful tool for diagonalizing a given Hamiltonian and is widely used in various branches of quantum physics. Here, generalizing this method, a condition is derived, under which the corresponding flow of a quantum state becomes geodesic in a submanifold of the projective Hilbert space, independently of specific initial conditions. This implies the geometric optimality of the present method as an algorithm of generating stationary states. The result is...

Source: http://arxiv.org/abs/0806.4425v3

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Sep 19, 2013
09/13

by
Yuichi Itto; Sumiyoshi Abe

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Stationary photon-atom entanglement is discussed by applying the method of flow equation to the Jaynes-Cummings model. A nonlocal continuous unitary transformation is explicitly constructed and the associated positive operator-valued measures for the photons and atom are obtained. Then, flow of the entanglement entropy is analyzed. A comment is also made on implementing the unitary operation in the method of flow equation. This method may offer a new strategy for quantum-state engineering.

Source: http://arxiv.org/abs/0708.1361v1

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Sep 18, 2013
09/13

by
Yuki Sughiyama; Sumiyoshi Abe

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Generalizing a recent work [T. Taniguchi and E. G. D. Cohen, J. Stat. Phys. 126, 1 (2006)] that was based on the Onsager-Machlup theory, a nonlinear relaxation process is considered for a macroscopic thermodynamic quantity. It is found that the fluctuation theorem holds in the nonlinear nonequilibrium regime if the change of the entropy characterized by local equilibria is appropriately renormalized. The fluctuation theorem for the ordinary entropy change is recovered in the linear...

Source: http://arxiv.org/abs/0803.1429v2

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Sep 19, 2013
09/13

by
Sumiyoshi Abe; Stefan Thurner

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It is shown that the laws of thermodynamics are extremely robust under generalizations of the form of entropy. Using the Bregman-type relative entropy, the Clausius inequality is proved to be always valid. This implies that thermodynamics is highly universal and does not rule out consistent generalization of the maximum entropy method.

Source: http://arxiv.org/abs/0708.1616v1

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Sep 19, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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Discovery of a new law for three dimensional spatial distance between the foci of successive earthquakes is reported. Analyzing the seismic data taken between 1984 and 2001 in southern California, it is found that the cumulative distribution of the distances follows the modified Zipf-Mandelbrot law, showing complexity of geometry of the events.

Source: http://arxiv.org/abs/cond-mat/0209567v2

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Sep 21, 2013
09/13

by
Sumiyoshi Abe; Norikazu Suzuki

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Discovery of a new scale-free law of earthquake phenomenon is reported. It is relevant to the structural and dynamical properties of the earthquake network proposed in a recent work [S. Abe and N. Suzuki, Europhys. Lett. 65, 581 (2004)]. The seismic data taken in southern California are mapped to an evolving directed network. It is found that statistics of the degrees of separation between two vertices is characterized by the Zipf-Mandelbrot distribution. This shows how a given earthquake can...

Source: http://arxiv.org/abs/cond-mat/0402226v1

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Sep 22, 2013
09/13

by
Sumiyoshi Abe; A. K. Rajagopal

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A definition of the nonadditive (nonextensive) conditional entropy indexed by q is presented. Based on the composition law in terms of it, the Shannon-Khinchin axioms are generalized and the uniqueness theorem is established for the Tsallis entropy. The nonadditive conditional entropy, when considered in the quantum context, is always positive for separable states but takes negative values for entangled states, indicating its utility for characterizing entanglement. A criterion deduced from it...

Source: http://arxiv.org/abs/quant-ph/0003145v1

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Sep 22, 2013
09/13

by
Sumiyoshi Abe; A. K. Rajagopal

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A self-consistent thermodynamic framework is presented for power-law canonical distributions based on the generalized central limit theorem by extending the discussion given by Khinchin for deriving Gibbsian canonical ensemble theory. The thermodynamic Legendre transform structure is invoked in establishing its connection to nonextensive statistical mechanics.

Source: http://arxiv.org/abs/cond-mat/0003380v2

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Sep 19, 2013
09/13

by
Sumiyoshi Abe; A. K. Rajagopal

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The second law of thermodynamics in nonextensive statistical mechanics is discussed in the quantum regime. Making use of the convexity property of the generalized relative entropy associated with the Tsallis entropy indexed by q, Clausius' inequality is shown to hold in the range of q between zero and two. This restriction on the range of the entropic index, q, is purely quantum mechanical and there exists no upper bound of q for validity of the second law in classical theory.

Source: http://arxiv.org/abs/cond-mat/0304066v1

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Sep 22, 2013
09/13

by
A. K. Rajagopal; Sumiyoshi Abe

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Traditionally the exponential canonical distributions of Gibbsian statistical mechanics are given theoretical justification in at least four different ways: steepest descent method, counting method, Khinchin's method based on te central limit theorem, and maximum entropy principle of Jaynes. Equally ubiquitous power-law canonical distributions are shown to be given similar justification by appropriately adopting these formulations.

Source: http://arxiv.org/abs/cond-mat/0003493v1

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Sep 22, 2013
09/13

by
Sumiyoshi Abe; A. K. Rajagopal

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It is shown that the distribution derived from the principle of maximum Tsallis entropy is a superposable Levy-type distribution. Concomitantly, the leading order correction to the limit distribution is also deduced. This demonstration fills an important gap in the derivation of the Levy-stable distribution from the nonextensive statistical framework.

Source: http://arxiv.org/abs/cond-mat/0003303v1

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Sep 22, 2013
09/13

by
A. K. Rajagopal; Sumiyoshi Abe

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Probability distributions defined on the half space are known to be quite different from those in the full space. Here, a nonextensive entropic treatment is presented for the half space in an analytic and self-consistent way. In this development, the ordinary first moment of the random variable X is divergent in contrast to the case of the full space. A general (nu)-th moment of X is considered as a constraint in the principle of maximum Tsallis entropy. The infinite divisibility of the...

Source: http://arxiv.org/abs/cond-mat/0003304v2

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Sep 23, 2013
09/13

by
A. K. Rajagopal; Sumiyoshi Abe

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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 to so as to preserve the concavity of the entropies. The Tsallis entropy is thereby found to be appropriately renormalized.

Source: http://arxiv.org/abs/quant-ph/9904029v1

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Sep 19, 2013
09/13

by
A. K. Rajagopal; Sumiyoshi Abe

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The foundations of the Boltzmann-Gibbs (BG) distributions for describing equilibrium statistical mechanics of systems are examined. Broadly, they fall into: (i) probabilistic paaroaches based on the principle of equal a priori probability (counting technique and method of steepest descents), law of large numbers, or the state density considerations and (ii) a variational scheme -- maximum entropy principle (due to Gibbs and Jaynes) subject to certain constraints. A minimum set of requirements...

Source: http://arxiv.org/abs/cond-mat/0303064v1

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Sep 20, 2013
09/13

by
Sumiyoshi Abe; G. B. Bagci

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In nonextensive statistical mechanics, two kinds of definitions have been considered for expectation valu of a physical quantity: one is the ordinary definition and the other is the normalized q-expectation value employing the escort distribution. Since both of them lead to the maximum-Tsallis-entropy distributions of a similar type, it is of crucial importance to determine which the correct physical one is. A point is that the definition of expectation value is indivisibly connected to the...

Source: http://arxiv.org/abs/cond-mat/0404253v1

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Jul 20, 2013
07/13

by
Sumiyoshi Abe; A. K. Rajagopal

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The Levy-type distributions are derived using the principle of maximum Tsallis nonextensive entropy both in the full and half spaces. The rates of convergence to the exact Levy stable distributions are determined by taking the N-fold convolutions of these distributions. The marked difference between the problems in the full and half spaces is elucidated analytically. It is found that the rates of convergence depend on the ranges of the Levy indices. An important result emerging from the present...

Source: http://arxiv.org/abs/cond-mat/0009399v1

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Jul 20, 2013
07/13

by
Sumiyoshi Abe; A. K. Rajagopal

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Macroscopic thermodynamics of equilibrium is constructed for systems obeying power-law canonical distributions. With this, the connection between macroscopic thermodynamics and microscopic statistical thermodynamics is generalized. This is complementary to the Gibbs theorem for the celebrated exponential canonical distributions of systems in contact with a heat bath. Thereby, a thermodynamic basis is provided for power-law phenomena ubiquitous in nature.

Source: http://arxiv.org/abs/cond-mat/0009400v1

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Sep 18, 2013
09/13

by
Sumiyoshi Abe; A. K. Rajagopal

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The forms of Euler and Gibbs-Duhem relations discussed in thermodynamics of extensive systems are shown to hold also for nonextensive systems with long-range interactions with a novel interpretation of entities appearing therein. In this way, the principles underlying Tsallis' scaling relations in equilibrium nonextensive thermostatistics are clarified.

Source: http://arxiv.org/abs/cond-mat/0401342v2

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Sep 19, 2013
09/13

by
Sumiyoshi Abe; A. K. Rajagopal

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Given physical systems, counting rule for their statistical mechanical descriptions need not be unique, in general. It is shown that this nonuniqueness leads to the existence of various canonical ensemble theories which equally arise from the definite microcanonical basis. Thus, the Gibbs theorem for canonical ensemble theory is not universal, and the maximum entropy principle is to be appropriately modefied for each physical context.

Source: http://arxiv.org/abs/quant-ph/9911097v2

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Sep 22, 2013
09/13

by
Sumiyoshi Abe; A. K. Rajagopal

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Based on the form invariance of the structures given by Khinchin's axiomatic foundations of information theory and the pseudoadditivity of the Tsallis entropy indexed by q, the concept of conditional entropy is generalized to the case of nonadditive (nonextensive) composite systems. The proposed nonadditive conditional entropy is classically nonnegative but can be negative in the quantum context, indicating its utility for characterizing quantum entanglement. A criterion deduced from it for...

Source: http://arxiv.org/abs/quant-ph/0001085v1

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Sep 23, 2013
09/13

by
Sumiyoshi Abe; A. K. Rajagopal

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The problem of quantum state inference and the concept of quantum entanglement are studied using a non-additive measure in the form of Tsallis entropy indexed by the positive parameter q. The maximum entropy principle associated with this entropy along with its thermodynamic interpretation are discussed in detail for the Einstein-Podolosky-Rosen pair of two spin-1/2 particles. Given the data on the Bell-Clauser-Horne-Shimony-Holt observable, the analytic expression is given for the inferred...

Source: http://arxiv.org/abs/quant-ph/9904088v1

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Sep 18, 2013
09/13

by
Sumiyoshi Abe; and A. K. Rajagopal

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Starting from microcanonical basis with the principle of equal a priori probability, it is found that, besides ordinary Boltzmann-Gibbs theory with the exponential distribution, a theory describing systems with power-law distributions can also be derived.

Source: http://arxiv.org/abs/cond-mat/0002159v1

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Sep 18, 2013
09/13

by
Sumiyoshi Abe; G. Kaniadakis; A. M. Scarfone

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The generalized entropic measure, which is optimized by a given arbitrary distribution under the constraints on normalization of the distribution and the finite ordinary expectation value of a physical random quantity, is considered and its Lesche stability property (that is different from thermodynamic stability) is examined. A general condition, under which the generalized entropy becomes stable, is derived. Examples known in the literature, including the entropy for the stretched-exponential...

Source: http://arxiv.org/abs/cond-mat/0401290v1