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From Newton’s Law to the Linear Boltzmann Equation Without Cut-Off

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Abstract

We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann–Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford’s strategy. Our proof relies then on a combination of Lanford’s strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.

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Authors and Affiliations

  1. UMR CNRS-UNS 7351, University of Nice Sophia-Antipolis, Parc Valrose, 06108, Nice Cedex 2, France

    Nathalie Ayi

  2. Project COFFEE, INRIA Sophia Antipolis Méditerranée, Valbonne, France

    Nathalie Ayi

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  1. Nathalie Ayi
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Correspondence to Nathalie Ayi.

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Communicated by C. Mouhot

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Ayi, N. From Newton’s Law to the Linear Boltzmann Equation Without Cut-Off. Commun. Math. Phys. 350, 1219–1274 (2017). https://doi.org/10.1007/s00220-016-2821-6

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  • DOI: https://doi.org/10.1007/s00220-016-2821-6

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