A variational principle for the specific entropy for symbolic systems with uncountable alphabets

Aguiar, D.; Cioletti, L.; Ruviaro, Ricardo

doi:10.1002/mana.201700229

Cited by 17 publications

(33 citation statements)

References 17 publications

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“…The proof of this claim follows the same lines of the proof of Theorem 3 in [1] (see also [10] chap. 1).…”

Section: Entropy and Information Gain For Compact Metric Spacesmentioning

confidence: 65%

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

Lopes¹,

Mengue²

2020

Preprint

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It is well known that in Information Theory and in Machine Learning the Kullback-Leibler divergence, which extends the concept of Shannon entropy, plays a fundamental role. Given an a priori probability ν on the measurable space X and a probability π on the measurable space X × Y we are able to define the entropy of π relative to ν. Using this entropy we obtain a natural definition of information gain for general measurable spaces which coincides with the mutual information given from the K-L divergence. There is a natural way of extending the concept of information gain by considering the entropy relative to transverse functions. This extends the meaning of specific information gain of Ergodic Theory. This will be also used to extend the meaning of specific information gain for the the XY model. Finally, we extend the concept of dynamical entropy production for the XY model. In this case we notice that the involution kernel is a natural tool to better understand entropy production. If the potential is symmetric the associated equilibrium probability has zero entropy production. The introduction of the a priori probability (indispensable here) and transverse functions is a novelty in the theory.

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“…The proof of this claim follows the same lines of the proof of Theorem 3 in [1] (see also [10] chap. 1).…”

Section: Entropy and Information Gain For Compact Metric Spacesmentioning

confidence: 65%

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

Lopes¹,

Mengue²

2020

Preprint

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“…Uniqueness. As far as we know, the first paper proving the uniqueness of equilibrium states for Hölder potentials in an uncountable alphabet setting is [ACR18]. The techniques employed there are no longer applicable here, because they are strongly dependent on the denseness of the Hölder potentials in the space C b (X, R), which may not be true if X is not compact.…”

Section: Discussionmentioning

confidence: 99%

Thermodynamic formalism for topological Markov chains on standard Borel spaces

Cioletti¹,

Silva²,

Stadlbauer³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space X ≡ E N , where E is a general standard Borel space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on X and obtain the existence of equilibrium states as finitely additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states does not have a purely finite additive part.We then apply our results to the construction of invariant measures of timehomogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite-dimensional diffusion.2010 Mathematics Subject Classification. 37D35, 28Dxx.

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“…For general properties of the Kullback-Leibler divergence in the dynamic setting see for instance [19] or [41] (when the alphabet is a compact metric space). The work [9] describes the relation of Kullback-Leibler divergence and [42] with the classical concept of specific entropy in Statistical Mechanics as presented in [31] (see also Proposition 40 in [41]).…”

Section: Preliminaries On Kl Divergencementioning

confidence: 99%

Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry

Lopes¹,

Ruggiero²

2021

Preprint

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In Nonequilibrium Thermodynamics and Information Theory, the relative entropy (or, KL divergence) plays a very important role. Consider a Hölder Jacobian J and the Ruelle (transfer) operator L log J . Two equilibrium probabilities µ 1 and µ 2 , can interact via a discretetime Thermodynamic Operation given by the action of the dual of the Ruelle operator L * log J . We argue that the law µ → L * log J (µ), producing nonequilibrium, can be seen as a Thermodynamic Operation after showing that it's a manifestation of the Second Law of Thermodynamics. We also show that the change of relative entropy satisfiesFurthermore, we describe sufficient conditions on J, µ 1 for getting h(L * log J (µ 1 )) ≥ h(µ 1 ), where h is entropy. Recalling a natural Riemannian metric in the Banach manifold of Hölder equilibrium probabilities we exhibit the second-order Taylor formula for an infinitesimal tangent change of KL divergence; a crucial estimate in Information Geometry. We introduce concepts like heat, work, volume, pressure, and internal energy, which play here the role of the analogous ones in Thermodynamics of gases. We briefly describe the MaxEnt method.

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A variational principle for the specific entropy for symbolic systems with uncountable alphabets

Abstract: In this paper we derived a variational principle for the specific entropy on the context of symbolic dynamics of compact metric space alphabets and use this result to obtain the uniqueness of the equilibrium states associated to a Walters potential.

Cited by 17 publications

References 17 publications

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel

Thermodynamic formalism for topological Markov chains on standard Borel spaces

Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry

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