Anomalous Energy Transport in FPU-\(\beta \) Chain
This paper is devoted to the derivation of a macroscopic fractional diffusion equation describing heat transport in an anharmonic chain. More precisely, we study here the so-called FPU-\(\beta \) chain, which is a very simple model for a one-dimensional crystal in which atoms are coupled to their nearest neighbors by a harmonic potential, weakly perturbed by a quartic potential. The starting point of our mathematical analysis is a kinetic equation: Lattice vibrations, responsible for heat transport, are modeled by an interacting gas of phonons whose evolution is described by the Boltzmann phonon equation. Our main result is the rigorous derivation of an anomalous diffusion equation starting from the linearized Boltzmann phonon equation.
KeywordsFPU-\(\beta \) chain Boltzmann phonon equation Hydrodynamic limit Anomalous diffusion
This material is based upon work supported by the Kinetic Research Network (KI-Net) under the NSF Grant No. RNMS #1107444. A.M. is partially supported by NSF Grant DMS-1201426. S.M thanks the University of Maryland and CSCAMM (Center for Scientific Computation and Mathematical Modeling) for their hospitality; the Cambridge Philosophical Society and Lucy Cavendish College (University of Cambridge) for their financial support. Thanks to Clément Mouhot from the University of Cambridge for his help and support. Thanks to Herbert Spohn from Technische Universität München for useful discussions. S.M is supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.
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