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PreprintCC BY 4.0HAL

THE HYDRODYNAMIC LIMIT FOR THE ENERGY TRANSPORT IN A STOCHASTICALLY PERTURBED HARMONIC CHAIN OF OSCILLATORS

  • Tomasz Komorowski
  • Stefano Olla
  • Marielle Simon
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Abstract

We consider a pinned harmonic chain perturbed by random velocity flips, which conserve the total energy. Previous results have established the diffusive hydrodynamic limit for the energy profile at the level of expectations. In this article, under suitable fourth-moment bounds on the initial data, we prove for the first time a law of large numbers for the empirical energy distribution. Our approach is based on the Wigner distribution and shows that its random fluctuations vanish in the macroscopic limit. Consequently, the empirical energy profile converges in probability to the deterministic solution of the corresponding heat equation.

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Cite

BibTeXRIS
BibTeX
@misc{komorowski2026hydrodynamic,
  title = {{THE HYDRODYNAMIC LIMIT FOR THE ENERGY TRANSPORT IN A STOCHASTICALLY PERTURBED HARMONIC CHAIN OF OSCILLATORS}},
  author = {Tomasz Komorowski and Stefano Olla and Marielle Simon},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11229},
  eprint = {2609.11229},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05741548v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - THE HYDRODYNAMIC LIMIT FOR THE ENERGY TRANSPORT IN A STOCHASTICALLY PERTURBED HARMONIC CHAIN OF OSCILLATORS
AU  - Tomasz Komorowski
AU  - Stefano Olla
AU  - Marielle Simon
PY  - 2026
DA  - 2026/09/06
PB  - HAL
DO  - 10.48550/arxiv.2609.11229
UR  - https://hal.science/hal-05741548v1
AB  - We consider a pinned harmonic chain perturbed by random velocity flips, which conserve the total energy. Previous results have established the diffusive hydrodynamic limit for the energy profile at the level of expectations. In this article, under suitable fourth-moment bounds on the initial data, we prove for the first time a law of large numbers for the empirical energy distribution. Our approach is based on the Wigner distribution and shows that its random fluctuations vanish in the macroscopic limit. Consequently, the empirical energy profile converges in probability to the deterministic solution of the corresponding heat equation.
ER  -