Diffusive Propagation of Energy in a Non-acoustic Chain

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.