Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential
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Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential Cédric Bernardin · François Huveneers
Received: 12 March 2012 / Accepted: 27 September 2012 © Springer-Verlag Berlin Heidelberg 2012
Abstract We study the thermal properties of a pinned disordered harmonic chain weakly perturbed by a noise and an anharmonic potential. The noise is controlled by a parameter λ → 0, and the anharmonicity by a parameter λ ≤ λ. Let κ be the conductivity of the chain, defined through the Green–Kubo formula. Under suitable hypotheses, we show that κ = O(λ) and, in the absence of anharmonic potential, that κ ∼ λ. This is in sharp contrast with the ordered chain for which κ ∼ 1/λ, and so shows the persistence of localization effects for a non-integrable dynamics. Mathematics Subject Classification 82C70 Transport processes · 82C44 Dynamics of disordered systems (random Ising systems, etc.) · 60H25 Random operators and equations 1 Introduction The mathematically rigorous derivation of macroscopic thermal properties of solids, starting from their microscopic description, is a serious challenge [8,17]. On the one hand, numerous experiments and numerical simulations show that, for a wide variety of materials, the heat flux is related to the gradient of temperature through a simple relation known as Fourier’s law: J = −κ(T ) ∇T, C. Bernardin Université de Lyon and CNRS, UMPA, UMR-CNRS 5669, ENS-Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07, France e-mail: [email protected] F. Huveneers (B) CEREMADE, Université de Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France e-mail: [email protected]
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C. Bernardin, F. Huveneers
where κ(T ) is the thermal conductivity of the solid. On the other hand, the mathematical understanding of this phenomenological law from the point of view of statistical mechanics is still lacking. A one-dimensional solid can be modelled by a chain of oscillators, each of them being possibly pinned by an external potential, and interacting through a nearest neighbour coupling. The case of homogeneous harmonic interactions can be readily analysed, but it has been realized that this very idealized solid behaves like a perfect conductor, and so violates Fourier’s law [21]. To take into account the physical observations, it is thus needed to consider more elaborate models, where ballistic transport of energy is broken. Here are two possible directions. On the one hand, adding some anharmonic interactions can drastically affect the conductivity of the chain [2,19]. Unfortunately, the rigorous study of anharmonic chains is in general out of reach, and even numerical simulations do not lead to completely unambiguous conclusions. In order to draw some clear picture, anharmonic interactions are mimicked in [3,6] by a stochastic noise that preserves total energy and possibly total momentum. The thermal behaviour of anharmonic solids is, at a qualitative level, correctly reproduced by this partially stochastic model. By instance, the
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