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Diffusion Without Spreading of a Wave Packet in Nonlinear Random Models

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Chaos, Fractals and Complexity (COSA-Net 2022)

Part of the book series: Springer Proceedings in Complexity ((SPCOM))

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Abstract

We discuss the long time behavior of a finite energy wave packet in nonlinear Hamiltonians on infinite lattices at arbitrary dimension, exhibiting linear Anderson localization. Strong arguments both mathematical and numerical, suggest for infinite models that small amplitude wave packets may generate stationary quasiperiodic solutions (KAM tori) almost indistinguishable from linear wave packets. The probability of this event is non vanishing at small enough amplitude and goes to unity at amplitude zero. Most other wave packets (non KAM tori) are chaotic. We discuss the Arnold diffusion conjecture (recently proven) and propose a modified Boltzmann statistics for wave packets valid in generic models. The consequence is that the probability that a chaotic wave packet spreads to zero amplitude is zero. It must always remain focused around one or few chaotic spots which wander randomly over the whole system and generate subdiffusion. In this paper, we study a class of so–called Ding Dong models, where the nonlinearities are replaced by hard core potentials, which also generate subdiffusion. We prove rigorously for these models that spreading is impossible for any initial wave packet.

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Authors and Affiliations

  1. Laboratoire Léon Brillouin, CEA Saclay, 91191, Gif-sur-Yvette, France

    Serge Aubry

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  1. Serge Aubry
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Correspondence to Serge Aubry .

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Editors and Affiliations

  1. Mathematics, University of Patras, Patras, Greece

    Tassos Bountis

  2. Geology and Geoenvironment, National and Kapodistrian University of Athens, Athens, Greece

    Filippos Vallianatos

  3. Institute of Nanoscience and Nanotechnology, National Center for Scientific Research Demokritos, Agia Paraskevi, Athens, Greece

    Astero Provata

  4. Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Dimitris Kugiumtzis

  5. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

    Yannis Kominis

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Aubry, S. (2023). Diffusion Without Spreading of a Wave Packet in Nonlinear Random Models. In: Bountis, T., Vallianatos, F., Provata, A., Kugiumtzis, D., Kominis, Y. (eds) Chaos, Fractals and Complexity. COSA-Net 2022. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-031-37404-3_1

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