A general two-scale criteria for logarithmic Sobolev inequalities

Lelièvre, Tony

doi:10.1016/j.jfa.2008.09.019

Cited by 17 publications

(31 citation statements)

References 7 publications

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“…Since µ ξ and the conditional measures µ Σz satisfy a logarithmic Sobolev inequality (see [H5] and [H2]), and under assumption [H3], we obtain that the measure µ also satisfies a logarithmic Sobolev inequality with some constant R > 0 (see [23]). Hence, by a computation similar to the one on E CG , we obtain…”

Section: Error Estimationmentioning

confidence: 92%

Effective dynamics using conditional expectations

2010

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The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable ξ(x), where x describes the configuration of the system in a high-dimensional space R n , and ξ is a smooth function with value in R (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on x ∈ R n , the equilibrium properties on ξ(x) are completely determined by the free energy. On the other hand, the question of the effective dynamics on ξ(x) is much more difficult to address. Starting from an overdamped Langevin equation on x ∈ R n , we propose an effective dynamics for ξ(x) ∈ R using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics.AMS classification scheme numbers: 35B40, 82C31, 60H10 ‡ In this article, we do not address the difficult question of how to find a good reaction coordinate. See for instance [24] for some discussion on that point.

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Section: Error Estimationmentioning

confidence: 92%

Effective dynamics using conditional expectations

2010

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“…, ρ I ). This result for a collection of I independent random variables admits generalizations when some correlations are introduced, see [87,41,63].…”

Section: Entropy Estimates: Basic Factsmentioning

confidence: 83%

Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics

Bris

Lelièvre

2012

Sci. China Math.

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“…We refer to [35] for a precise mathematical statement. The proof is based on entropy techniques [2], and the idea of two-scale analysis for logarithmic Sobolev inequalities [22,32].…”

Section: A First Example: the Adaptive Biasing Force Techniquementioning

confidence: 99%

Two Mathematical Tools to Analyze Metastable Stochastic Processes

Lelièvre

2012

Numerical Mathematics and Advanced Applications 2011

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We present how entropy estimates and logarithmic Sobolev inequalities on the one hand, and the notion of quasi-stationary distribution on the other hand, are useful tools to analyze metastable overdamped Langevin dynamics, in particular to quantify the degree of metastability. We discuss the interest of these approaches to estimate the efficiency of some classical algorithms used to speed up the sampling, and to evaluate the error introduced by some coarse-graining procedures. This paper is a summary of a plenary talk given by the author at the ENUMATH 2011 conference.

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A general two-scale criteria for logarithmic Sobolev inequalities

Cited by 17 publications

References 7 publications

Effective dynamics using conditional expectations

Effective dynamics using conditional expectations

Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics

Two Mathematical Tools to Analyze Metastable Stochastic Processes

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