On three new principles in non-equilibrium statistical mechanics and Markov semigroups of weak coupling limit type

Accardi, Luigi; Fagnola, Franco; Quezada, Roberto

doi:10.1142/s0219025716500090

Cited by 29 publications

(19 citation statements)

References 25 publications

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“…Let τ denote the normalized trace on A given by τ (A) = Tr[A]/Tr [1] for all A ∈ A. Then τ is a faithful state on A.…”

Section: Preliminary Definitions and Notationmentioning

confidence: 99%

“…Already in these papers, one finds several different, and nonequivalent, definitions. A number of more recent investigations [1,24,26,47,69] have added to the variety of meanings attached to this term. We therefore carefully explain the context of the definition that we use here, and why it is the most natural for our purposes.…”

Section: Detailed Balancementioning

confidence: 99%

“…Suppose that P t is completely positive for each t > 0. By (A.7), the GKS matrix of the identity transformation is c α,β (I) = δ α,(1,1) δ β, (1,1) .…”

Section: Theorem (Talagrand Type Inequality)mentioning

confidence: 99%

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Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

Cárlen

Maas

2017

Journal of Functional Analysis

126

250

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Abstract. We study a class of ergodic quantum Markov semigroups on finitedimensional unital C * -algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.

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“…Let τ denote the normalized trace on A given by τ (A) = Tr[A]/Tr [1] for all A ∈ A. Then τ is a faithful state on A.…”

Section: Preliminary Definitions and Notationmentioning

confidence: 99%

Section: Detailed Balancementioning

confidence: 99%

See 1 more Smart Citation

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

Cárlen

Maas

2017

Journal of Functional Analysis

126

250

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“…We consider the formal Lindblad generator (1) which is of weak coupling limit type (see Refs. [3,5]) since it arises in the weak coupling limit of a system with Hilbert space ℓ 2 (N) and Hamiltonian H S given by the number operator N coupled to a Boson reservoir in equilibrium with inverse temperature β > 0 and interaction operator…”

Section: The Modelmentioning

confidence: 99%

Quadratic open quantum harmonic oscillator

2020

Self Cite

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We study the quantum open system evolution described by a Gorini-Kossakowski-Sudarshan-Lindblad generator with creation and annihilation operators arising in Fock representations of the sl 2 Lie algebra. We show that any initial density matrix evolves to a fully supported density matrix and converges towards a unique equilibrium state. We show that the convergence is exponentially fast and we exactly compute the rate for a wide range of parameters. We also discuss the connection with the two-photon absorption and emission process.

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“…In the past four decades, general QMS have been studied extensively and many deep results have been obtained (see, e.g. [1,4,5,7,9,11] and references therein). One typical result in this aspect is the theorem established by Gorini, Kossakowski, Sudershan and Lindblad, which gives a characterization of the generater of a uniformly continuous QMS (see [15] for details).…”

Section: Introductionmentioning

confidence: 99%

Weighted number operators on Bernoulli functionals and quantum exclusion semigroups

Wang

Tang

Ren

2019

Journal of Mathematical Physics

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Quantum Bernoulli noises (QBN, for short) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anticommutation relation (CAR) in equal-time. In this paper, by using QBN, we first introduce a class of self-adjoint operators acting on Bernoulli functionals, which we call the weighted number operators. We then make clear spectral decompositions of these operators, and establish their commutation relations with the annihilation as well as the creation operators. We also obtain a necessary and sufficient condition for a weighted number operator to be bounded. Finally, as application of the above results, we construct a class of quantum Markov semigroups associated with the weighted number operators, which belong to the category of quantum exclusion semigroups. Some basic properties are shown of these quantum Markov semigroups, and examples are also given.

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On three new principles in non-equilibrium statistical mechanics and Markov semigroups of weak coupling limit type

Abstract: We introduce three new principles: the nonlinear Boltzmann-Gibbs prescription, the local KMS condition and the generalized detailed balance (GDB) condition. We prove the equivalence of the first two under general conditions and we discuss a master equation formulation of the third one

Cited by 29 publications

References 25 publications

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

Quadratic open quantum harmonic oscillator

Weighted number operators on Bernoulli functionals and quantum exclusion semigroups

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