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Nonequilibrium Phenomena in Nonlinear Lattices: From Slow Relaxation to Anomalous Transport

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Emerging Frontiers in Nonlinear Science

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 32))

Abstract

This chapter contains an overview of the effects of nonlinear interactions in selected problems of non-equilibrium statistical mechanics. Most of the emphasis is put on open setups, where energy is exchanged with the environment. With reference to a few models of classical coupled anharmonic oscillators, we review anomalous but general properties such as extremely slow relaxation processes, or non-Fourier heat transport.

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Notes

  1. 1.

    The actual implementation of a reservoir for the DNLS is less straightforward than for usual oscillator models [28]. Two main strategies have been proposed: the first is a Monte-Carlo dynamics [27] whereby the reservoir performs random perturbations \(\delta z_1\) of, say, the state variable \(z_1\) that are accepted or rejected according to a grand-canonical Metropolis cost-function \(\exp {[-\beta (\Delta H-\mu \Delta A)]}\), where \(\Delta H\) and \(\Delta A\) are respectively the variations of energy and mass produced by \(\delta z_1\). Between successive interactions with the environment the dynamics is Hamiltonian and can be integrated by symplectic algorithms [29]. Another approach is based on a Langevin dynamics with a dissipation designed in such a way that equilibrium corresponds to the grand-canonical measure [24].

  2. 2.

    Bounded fluctuations would be possible only in the presence of an attractor, but this is a Hamiltonian system.

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Acknowledgements

We thank L. Chirondojan and G.-L. Oppo for useful discussions. We acknowledge support from the project MIUR-PRIN2017 Coarse-grained description for non-equilibrium systems and transport phenomena (CO-NEST) n. 201798CZL.

Author information

Authors and Affiliations

  1. Dipartimento di Fisica e Astronomia, Università di Padova, via F. Marzolo 8 I-35131, Padova, Italy

    Stefano Iubini

  2. Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi via Madonna del Piano 10, I-50019, Sesto Fiorentino, Italy

    Stefano Iubini, Stefano Lepri, Roberto Livi & Paolo Politi

  3. Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1 I-50019, Sesto Fiorentino, Italy

    Stefano Lepri, Roberto Livi & Paolo Politi

  4. Dipartimento di Fisica e Astronomia, Università di Firenze, via G. Sansone 1 I-50019, Sesto Fiorentino, Italy

    Roberto Livi

  5. Institute for Pure and Applied Mathematics & SUPA University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom

    Antonio Politi

Authors
  1. Stefano Iubini
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  2. Stefano Lepri
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  3. Roberto Livi
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  5. Paolo Politi
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Corresponding author

Correspondence to Stefano Lepri .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA

    Panayotis G. Kevrekidis

  2. Grupo de Fisica No Lineal, Departamento de Fisica Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, Seville, Spain

    Jesús Cuevas-Maraver

  3. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Avadh Saxena

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Iubini, S., Lepri, S., Livi, R., Politi, A., Politi, P. (2020). Nonequilibrium Phenomena in Nonlinear Lattices: From Slow Relaxation to Anomalous Transport. In: Kevrekidis, P., Cuevas-Maraver, J., Saxena, A. (eds) Emerging Frontiers in Nonlinear Science. Nonlinear Systems and Complexity, vol 32. Springer, Cham. https://doi.org/10.1007/978-3-030-44992-6_8

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