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Superdiffusive Heat Transport in a Class of Deterministic One-dimensional Many-Particle Lorentz Gases

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Abstract

We study heat transport in a one-dimensional chain of a finite number N of identical cells, coupled at its boundaries to stochastic particle reservoirs. At the center of each cell, tracer particles collide with fixed scatterers, exchanging momentum. In a recent paper (Collet and Eckmann in Commun. Math. Phys. 287:1015, 2009), a spatially continuous version of this model was derived in a scaling regime where the scattering probability of the tracers is γ∼1/N, corresponding to the Grad limit. A Boltzmann-like equation describing the transport of heat was obtained. In this paper, we show numerically that the Boltzmann description obtained in Collet and Eckmann (Commun. Math. Phys. 287:1015, 2009) is indeed a bona fide limit of the particle model. Furthermore, we study the heat transport of the model when the scattering probability is 1, corresponding to deterministic dynamics. Thought as a lattice model in which particles jump between different scatterers the motion is persistent, with a persistence probability determined by the mass ratio among particles and scatterers, and a waiting time probability distribution with algebraic tails. We find that the heat and particle currents scale slower than 1/N, implying that this model exhibits anomalous heat and particle transport.

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Authors and Affiliations

  1. Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Pierre Collet

  2. Département de Physique Théorique, Université de Genève, Geneva, Switzerland

    Jean-Pierre Eckmann

  3. Section de Mathématiques, Université de Genève, Geneva, Switzerland

    Jean-Pierre Eckmann

  4. Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Sesto Fiorentino, Italy

    Carlos Mejía-Monasterio

Authors
  1. Pierre Collet
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  2. Jean-Pierre Eckmann
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  3. Carlos Mejía-Monasterio
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Correspondence to Jean-Pierre Eckmann.

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Collet, P., Eckmann, JP. & Mejía-Monasterio, C. Superdiffusive Heat Transport in a Class of Deterministic One-dimensional Many-Particle Lorentz Gases. J Stat Phys 136, 331–347 (2009). https://doi.org/10.1007/s10955-009-9783-4

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  • DOI: https://doi.org/10.1007/s10955-009-9783-4

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