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Category Theoretic Properties of the A. Rényi and C. Tsallis Entropies

Journal of Modern Physics 2016年第2期7卷 251-266页

作者： György Steinbrecher alberto sonnino Giorgio sonnino Physics Department Faculty of Science University of Craiova Craiova Romania Karlsruhe Institute of Technologies (KIT) Department of Electrical Engineering and Information Technologies Karlsruhe Germany Ecole Polytechnique de Louvain (EPL) Université Catholique de Louvain (UCL) Louvain-la-Neuve Belgium Department of Theoretical Physics and Mathematics Université Libre de Bruxelles (ULB) Brussels Belgium

The problem of embedding the Tsallis, Rényi and generalized Rényi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the Rényi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic partition functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original Rényi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and Rényi entropies. It is proved that in a similar manner, the partition functional that appears in the definition of the Generalized Rényi entropy is a multiplicative functional with respect to direct product and additive with respect to the disjoint sum, but its symmetry group is reduced compared to the case of classical Rényi entropy.

关键词： Rényi Entropy Generalized Rényi Entropy Measured Spaces Monoidal Category

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Determination of the Dynamic ITER Energy Confinement Time Scalings

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Journal of Modern Physics 2016年第12期7卷 1429-1439页

作者： Giorgio sonnino alberto sonnino Jarah Evslin Pasquale Nardone György Steinbrecher Department of Theoretical Physics and Mathematics Université Libre de Bruxelles (U.L.B.) Campus Plaine Brussels Belgium Royal Military School (RMS) Brussels Belgium Ecole Polytechnique de Louvain (EPL) Université Catholique de Louvain (UCL) Louvain-la-Neuve Belgium High Energy Nuclear Physics Group Institute of Modern Physics Chinese Academy of Sciences Lanzhou China Physics Department University of Craiova Craiova Romania

We derive the differential equation, which is satisfied by the ITER scalings for the dynamic energy confinement time. We show that this differential equation can also be obtained from the differential equation for the energy confinement time, derived from the energy balance equation, when the plasma is near the steady state. We find that the values of the scaling parameters are linked to the second derivative of the power loss, estimated at the steady state. As an example of an application, the solution of the differential equation for the energy confinement time is compared with the profile obtained by solving numerically the balance equations (closed by a transport model) for a concrete Tokamak-plasma.

关键词： Fusion Reactors Theory Design and Computerized Simulation

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