Abstract
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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Communicated by C. Le Bris
This paper has been partially supported by the ANR-15-CE40-0020-01 Grant LSD; T. Komorowski acknowledges the support of the Polish National Science Center Grant UMO-2012/07/B/ST1/03320.
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Komorowski, T., Olla, S. Diffusive Propagation of Energy in a Non-acoustic Chain. Arch Rational Mech Anal 223, 95–139 (2017). https://doi.org/10.1007/s00205-016-1032-9
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DOI: https://doi.org/10.1007/s00205-016-1032-9