Geodesic convexity of the relative entropy in reversible Markov chains

Mielke, Alexander

doi:10.1007/s00526-012-0538-8

Cited by 143 publications

(178 citation statements)

References 25 publications

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“…In the remainder of this section, we shall show that it is also quite fruitful in our finite-dimensional setting. Related work in the commutative setting can be found in [22,23,27,44,46,48]. For the difficulties relating to direct computation and analysis of the Hessian even for the Fermi Ornstein-Uhlenbeck semigroup, see our previous paper [11].…”

Section: Geodesic Convexity and Relaxation To Equilibriummentioning

confidence: 99%

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

Cárlen

Maas

2017

Journal of Functional Analysis

127

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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Section: Geodesic Convexity and Relaxation To Equilibriummentioning

confidence: 99%

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

Cárlen

Maas

2017

Journal of Functional Analysis

127

250

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“…see also [16]. Note that the vector field F (u) = K(u)DE(u) is nonlinear and that the matrices K(u) and M(u) have no homogeneity or concavity properties, in general.…”

Section: (A) Pure Reaction Systems and Markov Chainsmentioning

confidence: 99%

“…This includes the case of general reversible Markov chainsu = Qu, where Q ∈ R I×I is a stochastic generator (see also [15][16][17]). …”

Section: E(γ (S θmentioning

confidence: 99%

Gradient structures and geodesic convexity for reaction–diffusion systems

Liero

Mielke

2013

Phil. Trans. R. Soc. A.

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We consider systems of reaction–diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic λ -convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift–diffusion system, provide a survey on the applicability of the theory.

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“…with the diagonal semi-positive definite This establishes a gradient system ( E s , X s , K s ) for (3) in the sense of [2]. To verify the result, it is sufficient to calculate…”

Section: Fast Equilibria and Limit Systemmentioning

confidence: 99%

An entropic gradient structure for quasi‐steady‐state approximations of chemical reactions

Disser

2016

Proc Appl Math and Mech

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Passing to the limit of an infinite reaction rate in a slow-fast system of chemical reactions provides a quasi-steady state approximation (QSSA) of these systems. In case of reactions with detailed balance condition, this approximation includes a dimension reduction to a smaller state space. We show that the limit dynamics carry an entropic gradient structure on this smaller space. where R s , R f ∈ N are the numbers of fast and slow reactions, respectively, with R := R s + R f ≤ I. In general, subscripts or superscripts s, f, c denote slow, fast or conserved quantities and parameters of the system. Here, k r > 0 and κ r k r > 0 are the (unscaled) forward and backward reaction rates, , and γ r = α r − β r are the stoichiometric vectors. We write C r (c) := c αr − κ r c βr as a shorthand and assume that γ 1 , · · · , γ R are linearly independent and that all parameters are independent of ε. In particular, system (1) satisfies detailed balance, cf. [1,2].

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Geodesic convexity of the relative entropy in reversible Markov chains

Cited by 143 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

Gradient structures and geodesic convexity for reaction–diffusion systems

An entropic gradient structure for quasi‐steady‐state approximations of chemical reactions

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