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Finding the Jump Rate for Fastest Decay in the Goldstein–Taylor Model

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Abstract

This paper is about the rate of convergence to equilibrium for hypocoercive linear kinetic equations. We look for the spatially dependent jump rate which yields the fastest decay rate of perturbations. For the Goldstein–Taylor model, we show (i) that for a locally optimal jump rate the spectral bound is determined by multiple, possibly degenerate, eigenvectors and (ii) that globally the fastest decay is obtained with a spatially homogeneous jump rate. Our proofs rely on a connection to damped wave equations and a relationship to the spectral theory of Schrödinger operators.

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Notes

  1. One could also prove \(c=\int _0^{2\pi } (j_1(y)^2 - \rho _1(y)^2)\, \mathrm {d}y\).

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Acknowledgements

The authors gratefully acknowledge the support of the Hausdorff Research Institute for Mathematics (Bonn), through the Junior Trimester Program on Kinetic Theory. The authors thank Clément Mouhot, Laurent Desvillettes, Iván Moyano for the iscussions on this project.

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Authors and Affiliations

  1. Université de Paris and Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG), 75013, Paris, France

    Helge Dietert

  2. Institut für Mathematik, Universität Leipzig, 04103, Leipzig, Germany

    Helge Dietert

  3. Warwick Mathematics Institute, University of Warwick, Coventry, UK

    Josephine Evans

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  1. Helge Dietert
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Correspondence to Helge Dietert.

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Communicated by Eric A. Carlen.

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Dietert, H., Evans, J. Finding the Jump Rate for Fastest Decay in the Goldstein–Taylor Model. J Stat Phys 188, 1 (2022). https://doi.org/10.1007/s10955-022-02925-3

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