Upscaling from particle models to entropic gradient flows

Dirr, Nicolas; Laschos, Vaios; Zimmer, Johannes

doi:10.1063/1.4726509

Cited by 12 publications

(20 citation statements)

References 8 publications

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“…and these match one-to-one to the definition (24). This shows how the structure of the relative Fisher Information is to some extent 'built-in' in this system.…”

Section: Conclusion and Discussionsupporting

confidence: 64%

“…Gradient flows and large-deviation principles As mentioned in the introduction, this approach using the duality formulation of the rate functionals is motivated by our recent results on the connection between generalised gradient flows and large-deviation principles [2,3,24,26,27,52]. We want to discuss here how the two overlap but are not the same.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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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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“…and these match one-to-one to the definition (24). This shows how the structure of the relative Fisher Information is to some extent 'built-in' in this system.…”

Section: Conclusion and Discussionsupporting

confidence: 64%

Section: Conclusion and Discussionmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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“…Let α and β constant, then X n satisfies a large-deviations principle for n → ∞, with rate function J α,β given in (15). Moreover, the minimizer of J α,β satisfies the generalized gradient-flow equation (3).…”

Section: Statement 1 (Large Deviations)mentioning

confidence: 99%

“…Recently, stochasticity has also been shown to play a role in understanding the origin of various gradientflow systems, such as those with Wasserstein-type metrics [2,3,15,30,41]. In this paper, we ask the question whether these different roles of noise can be related:…”

mentioning

confidence: 99%

Quadratic and rate-independent limits for a large-deviations functional

Bonaschi

Peletier

2015

Continuum Mech. Thermodyn.

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We construct a stochastic model showing the relationship between noise, gradient flows and rateindependent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers' law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits, we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via 'L log L' gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.

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“…It is unclear to us whether these functionals are related to the large deviation functionals arising from some other microscopic dynamics. We also refer to [1,15] where the entropic gradient flow structure is explored on the basis of large deviation principles for linear diffusion equations on the real line (without boundary).…”

Section: 3mentioning

confidence: 99%

Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations

et al. 2014

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Abstract. We exhibit a large class of Lyapunov functionals for nonlinear drift-diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic many-particle systems, the zero range process and the Ginzburg-Landau dynamics, which we describe briefly. As an application, we prove linear inequalities between such an entropy-like functional and its entropy production functional for the boundary-driven porous medium equation in a bounded domain with positive Dirichlet conditions: this implies exponential rates of relaxation related to the first Dirichlet eigenvalue of the domain. We also derive Lyapunov functions for systems of nonlinear diffusion equations, and for nonlinear Markov processes with non-reversible stationary measures.

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Upscaling from particle models to entropic gradient flows

Cited by 12 publications

References 8 publications

Variational approach to coarse-graining of generalized gradient flows

Variational approach to coarse-graining of generalized gradient flows

Quadratic and rate-independent limits for a large-deviations functional

Lyapunov functionals for boundary-driven nonlinear drift–diffusion equations

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