19,635 research outputs found

    Geometry of escort distributions

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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 measures fluctuations of the crowding index of a multifractal. The Cramer-Rao inequality leads to the fundamental limit for precision of statistical estimate of the order of the escort distribution. It is also quantitatively discussed how inappropriate it is to use the original distribution instead of the escort distribution for calculating the expectation values of physical quantities in nonextensive statistical mechanics.Comment: 12 pages, no figure

    Macroscopic thermodynamics of equilibrium characterized by power-law canonical distributions

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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.Comment: 12 page

    Generalized entropies and the transformation group of superstatistics

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    Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures β\beta, so that the probability distribution is p(ϵi)0f(β)eβϵidβp(\epsilon_i) \propto \int_{0}^{\infty} f(\beta) e^{-\beta \epsilon_i}d\beta, where the `kernel' f(β)f(\beta) is nonnegative and normalized (f(β)dβ=1\int f(\beta)d \beta =1). We discuss the relation between this distribution and the generalized entropic form S=is(pi)S=\sum_i s(p_i). The first three Shannon-Khinchin axioms are assumed to hold. It then turns out that for a given distribution there are two different ways to construct the entropy. One approach uses escort probabilities and the other does not; the question of which to use must be decided empirically. The two approaches are related by a duality. The thermodynamic properties of the system can be quite different for the two approaches. In that connection we present the transformation laws for the superstatistical distributions under macroscopic state changes. The transformation group is the Euclidean group in one dimension.Comment: 5 pages, no figur

    Rapidity Gaps from Colour String Topologies

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    Diffractive deep inelastic scattering at HERA and diffractive W and jet production at the Tevatron are well described by soft colour exchange models. Their essence is the variation of colour string-field topologies giving both gap and no-gap events, with a smooth transition and thereby a unified description of all final states.Comment: 3 pages, 6 eps figures, contribution to the DIS 99 workshop proceedings, uses npb.st

    Macroscopic proof of the Jarzynski-Wojcik fluctuation theorem for heat exchange

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    In a recent work, Jarzynski and Wojcik (2004 Phys. Rev. Lett. 92, 230602) have shown by using the properties of Hamiltonian dynamics and a statistical mechanical consideration that, through contact, heat exchange between two systems initially prepared at different temperatures obeys a fluctuation theorem. Here, another proof is presented, in which only macroscopic thermodynamic quantities are employed. The detailed balance condition is found to play an essential role. As a result, the theorem is found to hold under very general conditions.Comment: 9 pages, 0 figure

    Nonadditive measure and quantum entanglement in a class of mixed states of N^n-system

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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 N^n-system, that is, an n-partite N-level system. It is shown how the strongest limitation on validity of local realism (i.e., separability of the state) can be obtained in a novel manner

    The Information Geometry of the One-Dimensional Potts Model

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    In various statistical-mechanical models the introduction of a metric onto the space of parameters (e.g. the temperature variable, β\beta, and the external field variable, hh, in the case of spin models) gives an alternative perspective on the phase structure. For the one-dimensional Ising model the scalar curvature, R{\cal R}, of this metric can be calculated explicitly in the thermodynamic limit and is found to be R=1+cosh(h)/sinh2(h)+exp(4β){\cal R} = 1 + \cosh (h) / \sqrt{\sinh^2 (h) + \exp (- 4 \beta)}. This is positive definite and, for physical fields and temperatures, diverges only at the zero-temperature, zero-field ``critical point'' of the model. In this note we calculate R{\cal R} for the one-dimensional qq-state Potts model, finding an expression of the form R=A(q,β,h)+B(q,β,h)/η(q,β,h){\cal R} = A(q,\beta,h) + B (q,\beta,h)/\sqrt{\eta(q,\beta,h)}, where η(q,β,h)\eta(q,\beta,h) is the Potts analogue of sinh2(h)+exp(4β)\sinh^2 (h) + \exp (- 4 \beta). This is no longer positive definite, but once again it diverges only at the critical point in the space of real parameters. We remark, however, that a naive analytic continuation to complex field reveals a further divergence in the Ising and Potts curvatures at the Lee-Yang edge.Comment: 9 pages + 4 eps figure

    Rigid rotators and diatomic molecules via Tsallis statistics

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    We obtain an analytic expression for the specific heat of a system of N rigid rotators exactly in the high temperature limit, and via a pertubative approach in the low temperature limit. We then evaluate the specific heat of a diatomic gas with both translational and rotational degrees of freedom, and conclude that there is a mixing between the translational and rotational degrees of freedom in nonextensive statistics.Comment: 12 page
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