Abstract
The thermal conductivity of insulating (dielectric) crystals is computed almost exclusively on the basis of the phonon Boltzmann equation. We refer to [1] for a discussion more complete than possible in this contribution. On the microscopic level the starting point is the Born-Oppenheimer approximation (see [2] for a modern version), which provides an effective Hamiltonian for the slow motion of the nuclei. Since their deviation from the equilibrium position is small, one is led to a wave equation with a weak nonlinearity. As already emphasized by R. Peierls in his seminal work [3], physically it is of importance to retain the structure resulting from the atomic lattice, which forces the discrete wave equation.
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References
H. Spohn, The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics, preprint.
S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, Lecture Notes in Mathematics 1821, Springer-Verlag, Berlin, Heidelberg, New York 2003.
R.E. Peierls, Zur kinetischen Theorie der Wärmeleitung in Kristallen, Annalen Physik 3, 1055–1101 (1929).
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L. Erdős, M. Salmhofer, and H.T. Yau, On the quantum Boltzmann equation, J. Stat. Phys. 116, 367–380 (2004).
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Spohn, H. (2006). Towards a Microscopic Derivation of the Phonon Boltzmann Equation. In: Asch, J., Joye, A. (eds) Mathematical Physics of Quantum Mechanics. Lecture Notes in Physics, vol 690. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-34273-7_21
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DOI: https://doi.org/10.1007/3-540-34273-7_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31026-6
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