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Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities

T.K. acknowledges the support of the Polish National Science Center grant DEC-2012/07/B/ST1/03320. The work of M.S. was supported by the ANR-14-CE25-0011 project (EDNHS) of the French National Research Agency (ANR), and by the Labex CEMPI (ANR-11-LABX-0007-01). S.O. has been partially supported by the ANR-15-CE40-0020-01 grant LSD. 6.
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  • We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random flip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation.

    Mathematics Subject Classification: 60K35, 74A25, 82C22.

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