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The emergence of the fractional heat equation from a one dimensional solid (FPU-$beta$ chain)

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Justifying mathematically Fourier law (how heat is transported from places of high temperature to places of low temperature) is one of the big challenges in Statistical Mechanics. But heat transfer is not only modelled by Fourier law; there is also the so-called ‘fractional’ Fourier law. We will focus on the latter.

A full justification of this law consists in deriving it from the atomic dynamics of the solid. This requires, in particular, a multiscale analysis, i.e., starting from the microscopic scale model (the one dimensional FPU -$beta$ chain representing a solid), we must reach the macroscopic model given by the fractional Fourier law (fractional heat equation). The strategy is to link these two models through an intermediate scale model (the kinetic Boltzmann-phonon equation) that serves as a bridge. I will comment on some work in progress with Professor Antoine Mellet, University of Maryland.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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