A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit

Grunewald, Natalie; Otto, Félix; Villani, Cédric; Westdickenberg, María G.

doi:10.1214/07-aihp200

Cited by 50 publications

(219 citation statements)

References 21 publications

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“…Existing quantitative methods such as [42,49] only work for gradient flows systems since they use crucially the gradient flow structures. The essential estimate that they need is the energy-dissipation inequality, which is similar to (4).…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

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Section: Conclusion and Discussionmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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“…[GOVW09]) and a transportation technique (cf. [CM10]) to give an alternative proof of the Eyring-Kramers formula.…”

Section: Results and Sketch Of Proofmentioning

confidence: 99%

“…We sketch a new approach to derive the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The new approach is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg [GOVW09] and of the mean-difference estimate introduced by Chafaï and Malrieu [CM10]. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the Gibbs measures restricted to a domain of attraction has a "good" Poincaré constant mimicking the fast convergence of the diffusion to metastable states.…”

Section: Introductionmentioning

confidence: 99%

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Some recent developments in functional inequalities

et al. 2014

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“…This is a classical problem in the theory of hydrodynamic limits, and various methods have been devised to tackle it. [28][29][30] In the context of a Ginzburg-Landau model, Guo, Papanicolaou, and Varadhan 28,31 show that a term of the form (23) can be replaced by a function of π N . More precisely, for ψ : T → R smooth and any > 0,…”

Section: The Replacement Lemmamentioning

confidence: 99%

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

Peletier

Redig

Vafayi

2014

Journal of Mathematical Physics

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We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter m, a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP(m) and the KMP, and a nonlinear heat equation for the GBEP(a). We prove the hydrodynamic limit rigorously for the BEP(m), and give a formal derivation for the GBEP(a).We then formally derive the pathwise large-deviation rate functional for the empirical measure of the three processes. These rate functionals imply gradient-flow structures for the limiting linear and nonlinear heat equations. We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form − log ρ; they involve dissipation or mobility terms of order ρ 2 for the linear heat equation, and a nonlinear function of ρ for the nonlinear heat equation.

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A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit

Cited by 50 publications

References 21 publications

Variational approach to coarse-graining of generalized gradient flows

Variational approach to coarse-graining of generalized gradient flows

Some recent developments in functional inequalities

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

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