Archive for Rational Mechanics and Analysis

, Volume 223, Issue 1, pp 95–139 | Cite as

Diffusive Propagation of Energy in a Non-acoustic Chain

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We consider a non-acoustic chain of harmonic oscillators with the dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. The macroscopic limits of the energy density, momentum and the curvature (or bending) of the chain satisfy a system of evolution equations. We prove that, in a diffusive space-time scaling, the curvature and momentum evolve following a linear system that corresponds to a damped Euler–Bernoulli beam equation. The macroscopic energy density evolves following a non linear diffusive equation. In particular, the energy transfer is diffusive in this dynamics. This provides a first rigorous example of a normal diffusion of energy in a one dimensional dynamics that conserves the momentum.


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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  2. 2.CNRS, CEREMADE, Université Paris Dauphine, PSL Research UniversityParisFrance

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