Entropy Production in Gaussian Thermostats

Dairbekov, N. S.; Paternain, Gabriel P.

doi:10.1007/s00220-006-0117-y

Cited by 16 publications

(52 citation statements)

References 22 publications

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“…For such a p and k = 1 it suffices to assume that φ is Anosov [6]. It is unknown if the Anosov hypothesis is enough for k ≥ 2.…”

Section: 6mentioning

confidence: 99%

“…For n = 2 this is proved in [6] but the proof requires to know apriori that φ is transitive, so both questions are intimately related.…”

Section: Final Remarks and Open Problemsmentioning

confidence: 99%

“…In this case one can consider the action of dφ on the bundle of Lagrangian subspaces and using the Maslov cycle, Proposition 5.1 and arguments similar to those in [23,Chapter 2] and [6], one can show that V is transversal to the weak bundles. (Details of this will appear elsewhere.)…”

Section: Final Remarks and Open Problemsmentioning

confidence: 99%

“…Like in [6] this non-perturbative result will be established by using Pestov type identities as in [4,8] for geodesic flows. A closely related result about the cohomological equation G E (u) = ϑ, where ϑ is an arbitrary 1-form is presented in Theorem 4.3.…”

Section: Introductionmentioning

confidence: 99%

“…In two degrees of freedom the problem was solved completely in [6]: an Anosov Gaussian thermostat has zero entropy production iff E has a global potential. The aim of the present paper is to provide similar results for n degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

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Entropy Production in Thermostats II

Dairbekov

Paternain

2007

J Stat Phys

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Abstract. We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field E has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfiesfor every 2-plane σ, where K w is the sectional curvature of the associated Weyl connection and E σ is the orthogonal projection of E onto σ. Then the entropy production of any SRB measure is positive unless E has a global potential. A related non-perturbative result is also obtained for certain generalized thermostats on surfaces.

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“…For such a p and k = 1 it suffices to assume that φ is Anosov [6]. It is unknown if the Anosov hypothesis is enough for k ≥ 2.…”

Section: 6mentioning

confidence: 99%

“…For n = 2 this is proved in [6] but the proof requires to know apriori that φ is transitive, so both questions are intimately related.…”

Section: Final Remarks and Open Problemsmentioning

confidence: 99%

Section: Final Remarks and Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Entropy Production in Thermostats II

Dairbekov

Paternain

2007

J Stat Phys

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The attenuated ray transform for connections and Higgs fields

2012

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We show that for a simple surface with boundary the attenuated ray transform in the presence of a unitary connection and a skew-Hermitian Higgs field is injective modulo the natural obstruction for functions and vector fields. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo a gauge transformation. The proofs involve a Pestov type energy identity for connections together with holomorphic gauge transformations which arrange the curvature of the connection to have definite sign.

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Holomorphic differentials, thermostats and Anosov flows

Mettler

Paternain

2018

Math. Ann.

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We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian two-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.

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Entropy Production in Gaussian Thermostats

Abstract: Abstract. We show that an arbitrary Anosov Gaussian thermostat on a surface is dissipative unless the external field has a global potential. This result is obtained by studying the cohomological equation of more general thermostats using the methods in [3].

Cited by 16 publications

References 22 publications

Entropy Production in Thermostats II

Entropy Production in Thermostats II

The attenuated ray transform for connections and Higgs fields

Holomorphic differentials, thermostats and Anosov flows

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