Kinetic theory of phonons in weakly anharmonic particle chains

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

Abstract

The aim of the chapter is to develop the kinetic theory of phonons in classical particle chains to a point which allows comparing the kinetic theory of normally conducting chains, with an anharmonic pinning potential, to the kinetic theory of the anomalously conducting FPU chains. In addition to reviewing the related literature, the chapter contains a streamlined derivation of the phonon Boltzmann collision operators using Wick polynomials, as well as details about the estimates which are needed to study the effect of the collision operator. This includes explicit solutions of the collisional constraints, both with and without harmonic pinning. We also recall in detail the derivation of the Green–Kubo formula for thermal conductivity in these systems, and the relation between entropy and the Boltzmann H-theorem for the phonon Boltzmann equations. The focus is in systems which are spatially translation invariant perturbations of thermal equilibrium states. We apply the results to obtain detailed predictions from kinetic theory for the Green–Kubo correlation functions, and hence the thermal conductivities, of the chain with a quartic pinning potential as well as the standard FPU-β chain.
Original languageEnglish
Title of host publicationThermal transport in low dimensions : from statistical physics to nanoscale heat transfer
EditorsStefano Lepri
Number of pages56
Volume921
PublisherSpringer
Publication date7 Apr 2016
Pages159-214
ISBN (Print)978-3-319-29259-5
ISBN (Electronic)978-3-319-29261-8
DOIs
Publication statusPublished - 7 Apr 2016
MoE publication typeA3 Book chapter

Publication series

NameLecture Notes in Physics
PublisherSpringer
Volume921
ISSN (Print)0075-8450

Fields of Science

  • 112 Statistics and probability

Cite this

Lukkarinen, J. (2016). Kinetic theory of phonons in weakly anharmonic particle chains. In S. Lepri (Ed.), Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer (Vol. 921, pp. 159-214). (Lecture Notes in Physics; Vol. 921). Springer. https://doi.org/10.1007/978-3-319-29261-8_4
Lukkarinen, Jani. / Kinetic theory of phonons in weakly anharmonic particle chains. Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer. editor / Stefano Lepri. Vol. 921 Springer, 2016. pp. 159-214 (Lecture Notes in Physics).
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Lukkarinen, J 2016, Kinetic theory of phonons in weakly anharmonic particle chains. in S Lepri (ed.), Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer. vol. 921, Lecture Notes in Physics, vol. 921, Springer, pp. 159-214. https://doi.org/10.1007/978-3-319-29261-8_4

Kinetic theory of phonons in weakly anharmonic particle chains. / Lukkarinen, Jani.

Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer. ed. / Stefano Lepri. Vol. 921 Springer, 2016. p. 159-214 (Lecture Notes in Physics; Vol. 921).

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

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AB - The aim of the chapter is to develop the kinetic theory of phonons in classical particle chains to a point which allows comparing the kinetic theory of normally conducting chains, with an anharmonic pinning potential, to the kinetic theory of the anomalously conducting FPU chains. In addition to reviewing the related literature, the chapter contains a streamlined derivation of the phonon Boltzmann collision operators using Wick polynomials, as well as details about the estimates which are needed to study the effect of the collision operator. This includes explicit solutions of the collisional constraints, both with and without harmonic pinning. We also recall in detail the derivation of the Green–Kubo formula for thermal conductivity in these systems, and the relation between entropy and the Boltzmann H-theorem for the phonon Boltzmann equations. The focus is in systems which are spatially translation invariant perturbations of thermal equilibrium states. We apply the results to obtain detailed predictions from kinetic theory for the Green–Kubo correlation functions, and hence the thermal conductivities, of the chain with a quartic pinning potential as well as the standard FPU-β chain.

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Lukkarinen J. Kinetic theory of phonons in weakly anharmonic particle chains. In Lepri S, editor, Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer. Vol. 921. Springer. 2016. p. 159-214. (Lecture Notes in Physics). https://doi.org/10.1007/978-3-319-29261-8_4