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Thermal Transport in Low Dimensions pp 1–37Cite as

Heat Transport in Low Dimensions: Introduction and Phenomenology

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Part of the book series: Lecture Notes in Physics ((LNP,volume 921))

Abstract

In this chapter we introduce some of the basic models and concepts that will be discussed throughout the volume. In particular we describe systems of nonlinear oscillators arranged on low-dimensional lattices and summarize the phenomenology of their transport properties.

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Notes

  1. 1.

    For historical reasons two of the scaling exponents introduced in this section are conventionally denoted by the same Greek letters, α and β, adopted for the FPU models described in Sect. 1.2.

  2. 2.

    From the mathematical point of view, the existence of a unique stationary measure is a relevant question and has been proven in some specific cases models of this class, see the review [8, 28, 29].

  3. 3.

    Temperature discontinuities may appear at the chain boundaries. This is a manifestation of the well-known Kapitza resistance, the temperature discontinuity arising when a heat flux is maintained across an interface among two substances. This discontinuity is the result of a boundary resistance, that is explained as a “phonon mismatch” between the two media: see [2] for a discussion of the class of models at hand.

  4. 4.

    In fact, the quadratic kernel corresponds to a quadratic force originating from the leading cubic nonlinearity of any asymmetric interaction potential, while a quartic leading nonlinearity of a symmetric interaction potential yields a cubic kernel (force).

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Acknowledgements

We wish to thank L. Delfini and S. Iubini for their effective contribution to the achievement of several results summarized in this chapter.

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Authors and Affiliations

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

    Stefano Lepri

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

    Roberto Livi

  3. Institute for Complex Systems and Mathematical Biology & SUPA University of Aberdeen, Aberdeen, AB24 3UE, UK

    Antonio Politi

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Lepri, S., Livi, R., Politi, A. (2016). Heat Transport in Low Dimensions: Introduction and Phenomenology. In: Lepri, S. (eds) Thermal Transport in Low Dimensions. Lecture Notes in Physics, vol 921. Springer, Cham. https://doi.org/10.1007/978-3-319-29261-8_1

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