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Toward the Fourier law for a weakly interacting anharmonic crystal

Authors: Carlangelo Liverani and Stefano Olla
Journal: J. Amer. Math. Soc. 25 (2012), 555-583
MSC (2010): Primary 82C70, 60F17, 80A20
Published electronically: December 1, 2011
MathSciNet review: 2869027
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Abstract: For a system of weakly interacting anharmonic oscillators, perturbed by an energy-preserving stochastic dynamics, we prove an autonomous (stochastic) evolution for the energies at large time scale (with respect to the coupling parameter). It turns out that this macroscopic evolution is given by the so-called conservative (nongradient) Ginzburg-Landau system of stochastic differential equations. The proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics.

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Additional Information

Carlangelo Liverani
Affiliation: Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy

Stefano Olla
Affiliation: CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, 75775 Paris-Cedex 16, France and INRIA - Université Paris Est, CERMICS, Projet MICMAC, Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée Cedex 2, France

Keywords: Weak coupling, scaling limits, hypoellipticity, hypocoercivity, Ginzburg-Landau dynamics, heat equation
Received by editor(s): January 18, 2011
Received by editor(s) in revised form: November 6, 2011
Published electronically: December 1, 2011
Additional Notes: This paper has been partially supported by the European Advanced Grant Macroscopic Laws and Dynamical Systems (MALADY) (ERC AdG 246953), by Agence Nationale de la Recherche, under grant ANR-07-BLAN-2-184264 (LHMSHE), and by MIUR under the grant PRIN 2007B3RBEY
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.