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Abstract: We consider the limit of a linear kinetic equation with a degenerate scattering kernel and a reflection-transmission-absorption condition at an interface. An equation of this type arises from the kinetic limit of a microscopic harmonic chain of oscillators whose dynamics is perturbed by a stochastic term, conserving energy and momentum. The chain is in contact, via one oscillator, with a heat bath, which, in the limit, generates the boundary condition at the
interface.
It is known that in the absence of the interface, the solution of the kinetic equation exhibits either superdiffusive, or diffusive behavior, in the proper long time - large scale limit, depending on the dispersion relation of the harmonic chain. We discuss how the presence of the interface influences the boundary condition for the limiting diffusion, or anomalous diffusion.
The presented results have been obtained in collaboration with G. Basile (Univ.
Roma I), A. Bobrowski (Lublin Univ. of Techn.), K. Bogdan (WrocÅ‚aw Univ. of Sci. and
Techn.), L. Arino (Ensta, Paris), S. Olla (Univ. Paris-Dauphine and GSSI, Lâ€™Aquila),
L. Ryzhik (Stanford Univ.), H. Spohn (TU, Munich).