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High Frequency Limit for a Chain of Harmonic Oscillators with a Point Langevin Thermostat

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Abstract

We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. In the high frequency limit, we establish the reflection-transmission coefficients for the wave energy for the scattering off the thermostat. To our surprise, even though the thermostat fluctuations are time-dependent, the scattering does not couple wave energy at various frequencies.

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Acknowledgements

TK was partially supported by the NCN Grant 2016/23/B/ST1/00492, SO by the French Agence Nationale Recherche Grant LSD ANR-15-CE40-0020-01, and LR by an NSF Grant DMS-1613603 and by ONR. This work was partially supported by the Grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. TK and LR thank Université Paris Dauphine for its hospitality during the time this work was completed.

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warsaw, Poland

    Tomasz Komorowski

  2. CEREMADE, UMR-CNRS, Université de Paris Dauphine, PSL Research University, Place du Maréchal De Lattre De Tassigny, 75016, Paris, France

    Stefano Olla

  3. Mathematics Department, Stanford University, Stanford, CA, 94305, USA

    Lenya Ryzhik

  4. Zentrum Mathematik and Physik Department, Technische Universität Munchen, Boltzmannstrasse 3, 85747, Munich, Germany

    Herbert Spohn

Authors
  1. Tomasz Komorowski
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  2. Stefano Olla
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  3. Lenya Ryzhik
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  4. Herbert Spohn
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Correspondence to Stefano Olla.

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Communicated by F. Otto

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Komorowski, T., Olla, S., Ryzhik, L. et al. High Frequency Limit for a Chain of Harmonic Oscillators with a Point Langevin Thermostat. Arch Rational Mech Anal 237, 497–543 (2020). https://doi.org/10.1007/s00205-020-01513-7

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  • DOI: https://doi.org/10.1007/s00205-020-01513-7

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