Duality between eigenfunctions and eigendistributions of Ruelle and Koopman operators via an integral kernel

Giulietti, Paolo; Lopes, Artur O.; Pit, Vincent

doi:10.1142/s021949371660011x

Cited by 7 publications

(8 citation statements)

References 16 publications

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“…Remark 28. Indeed, the claim follows from a simple reasoning using the involution kernel (see Definition 44 in Section 9) as described in [36]. Indeed, from the involution kernel and ρ one can get (see (112)) a positive eigenfunction associated to the Ruelle operator L log J 3 , but this impossible if λ = 1 (see Theorem 2.2 in [50] or Proposition 12 in [45]).…”

Section: Preliminaries On Kl Divergencementioning

confidence: 99%

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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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Section: Preliminaries On Kl Divergencementioning

confidence: 99%

“…is the main eigenfunction of the Ruelle operator L A . Other eigenfunctions (not strictly positive) can be eventually obtained via eigendistributions for L * A (see [36]). Email of Artur O. Lopes is arturoscar.lopes@gmail.com Email of R. Ruggiero is Rafael.O.Ruggiero@gmail.com…”

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confidence: 99%

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

Lopes¹,

Ruggiero²

2021

Preprint

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“…To simplify the notation we write simply A(x), A * (y) and W (y|x) during the computations. For general properties of involutions kernels, the reader can see the references [BLT06], [LMMS15] and [GLP16].…”

Section: Involution Kernel Representations Of Eigenfunctionsmentioning

confidence: 99%

Correlation inequalities and monotonicity properties of the Ruelle operator

Cioletti

Lopes

2019

Stoch. Dyn.

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Let X = {1, −1} N be the symbolic space endowed with a partial order , where x y, if x j ≥ y j , for all j ∈ N. A function f : X → R is called increasing if any pair x, y ∈ X, such that x y, we have f (x) ≥ f (y). A Borel probability measure µ over X is said to satisfy the FKG inequality if for any pair of continuous increasing functions f and g we have µ(f g) − µ(f )µ(g) ≥ 0. In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive specifications our proof has advantage of to be easily adapted for suitable subshifts. We review (and provide proofs in our setting) some classical results about the long-range Ising model on the lattice N and use them to deduce some monotonicity properties of the associated Ruelle operator and their relations with phase transitions.As is widely known, for some continuous potentials does not exists a positive continuous eigenfunction associated to the spectral radius of the Ruelle operator acting on C(X). Here we employed some ideas related to the involution kernel in order to solve the main eigenvalue problem in a suitable sense for a class of potentials having low regularity. From this we obtain an explicit tight upper bound for the main eigenvalue (consequently for the pressure) of the Ruelle operator associated to Ising models with 1/r 2+ε interaction energy. Extensions of the Ruelle operator to suitable Hilbert Spaces are considered and a theorem solving to the main eigenvalue problem (in a weak sense) is obtained by using the Lions-Lax-Milgram theorem. We generalize results due to P. Hulse on attractive g-measures.We also present the graph of the main eigenfunction in some examples -in some cases the numerical approximation shows the evidence of not being continuous.

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“…This is not possible due to properties of the involution kernel (see [39]). These two eigenprobabilities would determine via the involution kernel two positive eigenfunctions for another Ruelle operator L f * , with the eigenvalues λ 1 and λ 2 (see [25]), where f * is the dual potential for f . This is not possible (see for instance [41] or Proposition 1 in [39]).…”

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confidence: 99%

“…This is not possible (see for instance [41] or Proposition 1 in [39]). Anyway, eigendistributions for L * f may exist (see [25]).…”

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confidence: 99%

Phase Transitions in One-Dimensional Translation Invariant Systems: A Ruelle Operator Approach

Cioletti

Lopes

2015

J Stat Phys

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We consider a family of potentials f , derived from the Hofbauer potentials, on the symbolic space Ω = {0, 1} N and the shift mapping σ acting on it. A Ruelle operator framework is employed to show there is a phase transition when the temperature varies in the following senses: the pressure is not analytic, there are multiple eigenprobabilities for the dual of the Ruelle operator, the DLR-Gibbs measure is not unique and finally the Thermodynamic Limit is not unique. Additionally, we explicitly calculate the critical points for these phase transitions. Some examples which are not of Hofbauer type are also considered. The non-uniqueness of the Thermodynamic Limit is proved by considering a version of a Renewal Equation.We also show that the correlations decay polynomially and compute the decay ratio.MSC: 82B20, 82B41, 82B26, 37D35, 37A35, 37A50, 37A60.

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Duality between eigenfunctions and eigendistributions of Ruelle and Koopman operators via an integral kernel

Cited by 7 publications

References 16 publications

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

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

Correlation inequalities and monotonicity properties of the Ruelle operator

Phase Transitions in One-Dimensional Translation Invariant Systems: A Ruelle Operator Approach

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