Generalized extensive entropies for studying dynamical systems in highly anisotropic phase spaces

Sonnino, Giorgio; Steinbrecher, György

doi:10.1103/physreve.89.062106

Cited by 6 publications

(48 citation statements)

References 46 publications

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“…These relations give the geometrical interpretation of the generalized entropies (for further information Refs to [13]…”

Section: Definitionsmentioning

confidence: 99%

“…In analogy to the properties from Equations (24), (25) we have a perfect correspondence with the classical case [13]. Consider the case when the measured spaces, measures, densities entering in the definition of the GRE from Equations (42)-(46) are decomposed as follows…”

Section: The Additivity Of Gre Multiplicative Property Ofmentioning

confidence: 99%

“…The interest in the study of the generalizations of the Shannon entropy in the recent years is due to the multiple applications of the Tsallis and Rényi entropy or the associated Rényi divergence [7] [8] [11] [12]. We also mention that similar to the classical H theorem of L. Boltzmann, the generalizations of the Rényi entropy, as well as the original Rényi entropy, are Liapunov functions for a large class of stochastic processes described by generalized Fokker-Planck equations, more exactly by Fokker-Planck equation where the drift term and the diffusion tensor are itself dependent on some external random variable [13]. We mention that in the case of suitable singular limiting procedure, both the Tsallis and Rényi entropies give the same limit: the Shannon entropy.…”

Section: Introductionmentioning

confidence: 99%

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

Steinbrecher¹,

Sonnino²,

Sonnino³

2016

JMP

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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.

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“…These relations give the geometrical interpretation of the generalized entropies (for further information Refs to [13]…”

Section: Definitionsmentioning

confidence: 99%

Section: The Additivity Of Gre Multiplicative Property Ofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Steinbrecher¹,

Sonnino²,

Sonnino³

2016

JMP

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“…In particular, this expression is inadequate for treating dynamical systems in highly anisotropic phase space. In Sonnino and Steinbrecher (2014) we show that a large class of anisotropic dynamical systems subject to random perturbations, including particle transport in random media, can adequately be treated by assuming the validity of the MaxEnt principle of the generalized Rényi entropies. In these cases, these entropies play the role of Liapunov functionals (Sonnino and Steinbrecher 2014).…”

Section: Derivation Of a Stationary Distribution Function Recently Mmentioning

confidence: 99%

“…However, as shown in (Sonnino and Steinbrecher 2014), (2.3) is unable to treat dynamical systems in highly anisotropic phase space such as turbulent systems or anisotropic dynamical systems subject to random perturbations. Sonnino and Steinbrecher (2014) suggests that these cases may equally be treated by the information theory. The price to pay is that the MaxEnt principle should be applied to more sophisticated expressions of entropy functionals, such as the generalized Rényi entropies (GRE).…”

Section: Estimation Of the Free Parameters In The Ddf For Ohmic Fulmentioning

confidence: 99%

Stationary distribution functions for ohmic Tokamak-plasmas in the weak-collisional transport regime by MaxEnt principle

et al. 2014

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In previous works, we derived stationary density distribution functions (DDF) where the local equilibrium is determined by imposing the maximum entropy (MaxEnt) principle, under the scale invariance restrictions, and the minimum entropy production theorem.In this paper we demonstrate that it is possible to reobtain these DDF solely from the MaxEnt principle subject to suitable scale invariant restrictions in all the variables. For the sake of concreteness, we analyze the example of ohmic, fully ionized, tokamakplasmas, in the weak-collisional transport regime. In this case we show that it is possible to reinterpret the stationary distribution function in terms of the Prigogine distribution function where the logarithm of the DDF is directly linked to the entropy production of the plasma. This leads to the suggestive idea that also the stationary neoclassical distribution functions, for magnetically confined plasmas in the collisional transport regimes, may be derived solely by the MaxEnt principle.

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Category theoretic properties of the A. Rényi and C. Tsallis entropies

Steinbrecher,

Sonnino,

Sonnino

2015

Preprint

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The problem of embedding the Tsallis and 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 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.

show abstract

Generalized extensive entropies for studying dynamical systems in highly anisotropic phase spaces

Cited by 6 publications

References 46 publications

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

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

Stationary distribution functions for ohmic Tokamak-plasmas in the weak-collisional transport regime by MaxEnt principle

Category theoretic properties of the A. Rényi and C. Tsallis entropies

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Generalized extensive entropies for studying dynamical systems in highly anisotropic phase spaces

Cited by 6 publications

References 46 publications

Category Theoretic Properties of the A. R&#233;nyi and C. Tsallis Entropies

Category Theoretic Properties of the A. R&#233;nyi and C. Tsallis Entropies

Stationary distribution functions for ohmic Tokamak-plasmas in the weak-collisional transport regime by MaxEnt principle

Category theoretic properties of the A. Rényi and C. Tsallis entropies

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

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