The phonon boltzmann equation, Properties and link to weakly anharmonic lattice dynamics

Journal of Statistical Physics

Volumn 124, Issue 2-4, 2006, Pages 1041-1104

  Spohn, Herbert ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Feynman diagrams; H theorem; Kinetic limit; Transport equations; Wigner function

Indexed keywords

EID: 33750067455     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-005-8088-5     Document Type: Article

References (52)

1. R. E. Peierls, Zur kinetischen Theorie der Wärmeleitung in Kristallen. Annalen Physik 3:1055-1101 (1929).
2. G. Leibfried, Gittertheorie der mechanischen und thermischen Eigenschaften der Kristalle, Handbuch der Physik Band VII/1, Kristallphysik, ed. S. Flügge, Springer, Berlin 1955.
3. V. L. Gurevich, Transport in Phonon Systems, North-Holland 1986.
4. G. P. Srivastava, The Physics of Phonons, Adam Hilger, Bristol 1990.
5. J. Callaway, Quantum Theory of the Solid State, Academic Press 1974.
6. A. J. H. McGaughey and M. Kaviany, Quantitative validation of the Boltzmann transport equation phonon thermal conductivity model under the single mode relaxation time approximation. Phys. Rev. B69:094303 (2004).
7. C. J. Glassbrenner and G. Slack, Thermal conductivity of Silicon and Germanium from 3ÌK to the melting point. Phys. Rev. 134:A1058-A1069 (1964).
8. S. Fujita, Introduction to Nonequilibrium Quantum Statistical Mechanics, Saunders 1966.
9. L. Ryzhik, G. Papanicolaou, and J. B. Keller, Transport equations for elastic and other waves in random media. Wave Motion 24:327-370 (1996).
10. V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence: I Wave Turbulence. Springer, Berlin 1992.
11. P. Janssen, The Interaction of Waves and Wind, Cambridge University Press 2004.
12. G. Eyink and H. Spohn, Space-time invariant states of the ideal gas with finite number, energy, and entropy density, in: On Dobrushin's Way. From Probability Theory to Statistical Physics, eds. R. Minlos, S. Shlosman, and Y. Suchov, Advances in the Mathematical Sciences, Ser. 2, Vol. 198, pp. 71-89. AMS 2000.
13. A. Mielke, Macroscopic behavior of microscopic oscillations in harmonic lattices, preprint 118 of SPP 'Analysis, Modeling and Simulation of Multiscale Problems' (2004).
14. R. L. Dobrushin, A. Pellegrinotti, Yu.M. Suhov, and L. Triolo, One-dimensional harmonic lattice caricature of hydrodynamics. J. Stat. Phys. 43:571-607 (1986).
15. T. Dudnikova and H. Spohn, Local stationarity for lattice dynamics in the harmonic approximation, preprint, arXiv:math-ph/0505031.
16. L. Erdös, M. Salmhofer, and H. T. Yau, On the quantum Boltzmann equation. J. Stat. Phys. 116:367-380 (2004).
17. O. Bratelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2, 2nd edition. Springer, Berlin 1996.
18. H. Spohn, unpublished notes, 2005.
19. D. Benedetto, F. Castella, R. Esposito, and M. Pulvirenti, Some considerations on the derivation of the nonlinear quantum Boltzmann equation. J. Stat. Phys. 116:381-410 (2004).
20. N. M. Hugenholtz, Derivation of the Boltzmann equation for a Fermi gas. J. Stat. Phys. 32:231-254 (1983).
21. N. T. Ho and L. J. Landau, Fermi gas on a lattice in the van Hove limit. J. Stat. Phys. 87:821-845 (1997).
22. C. Cercignani and G. M. Kremer, On relativistic collisional invariants. J. Stat. Phys. 96:439-445 (1999).
23. J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, preprint, arXiv:math-ph/0505075.
24. L. Erdös and H. T. Yau, Linear Boltzmann equation as a weak coupling limit of the random Schrödinger equation. Commun. Pure Appl. Math. 53:667-735 (2000).
25. L. Erdös, Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field. J. Stat. Phys. 107:1043-1128 (2002).
26. T. Chen, Localization lengths and Boltzmann limit for the Anderson model at small disorder in dimension 3. J. Stat. Phys., online (2005).
27. r-Convergence of a random Schrödinger to a linear Boltzmann evolution, preprint, arXiv:math-ph/0407037.
28. L. W. Nordheim, On the kinetic method in the new statistics and its application in the electron theory of conductivity. Proc. Roy. Soc. 689-698 (1929).
29. R. Brout, I. Prigogine, Statistical mechanics of irreversible processes part V: anharmonic forces. Physica 22:35-47 (1956).
30. I. Prigogine, Nonequilibrium Statistical Mechanics, Wiley-Interscience, New York 1962.
31. M. Kac, Foundations of Kinetic Theory, Proc. 3rd Berkeley Symposium, Math. Stat. Prob., J. Newman, ed., Univ. of California, Vol. 3, pp. 174-197, 1956.
32. M. C. Carvalho, E. Carlen, and M. Loss, Determination of the spectral gap in Kac's master equation and related stochastic evolutions. Acta Math. 191:1-54 (2003).
33. L. van Hove, Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21: 517-540 (1955).
34. E. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 49:749-759 (1932).
35. L. Erdös, M. Salmhofer, H.-T. Yau, Quantum diffusion of random Schrödinger evolution in the scaling limit, preprint, arXiv:math-ph/ 0502025.
36. G. Bal, T. Komorowski, and L. Ryzhik, Self-averaging of Wigner transforms in random media. Comm. Math. Phys. 242: 81-135 (2003).
37. E. Fermi, J. Pasta, and S. Ulam, Studies in nonlinear problems, I, in Nonlinear Wave Motion, A.C. Newell, ed. (American Mathematical Society, Providence, RI, 1974), pp. 143-156. Originally published as Los Alamos Report LA-1940 in 1955.
38. E. A. Jackson, Nonlinearity and irreversibility in lattice dynamics, Rocky Mount. J. Math. 8:127-196 (1978).
39. S. Lepri, R. Livi, and A. Politi, Thermal conductivity in classical low-dimensional lattices. Physics Reports 377:1-80 (2003).
40. F. Bonetto, J. L. Lebowitz, and L. Rey-Bellet, Fourier's law: A challenge to theorists, in Mathematical Physics 2000, A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski, eds. (Imperial College Press, London, 2000), pp. 128-150.
41. 4 field theory. Ann. Phys. 295:50-80 (2002).
42. G. R. Lee-Dadswell, B. G. Nickel, and C. G. Gray, Thermal conductivity and bulk viscosity in quartic oscillator chains, preprint (2005).
43. S. Lepri, Memory effects and heat transport in one-dimensional insulators. Eur. Phys. J. B18:441-446 (2000).
44. R. Lefevere and A. Schenkel, Perturbative analysis of anharmonic chains of oscillators out of equilibrium. J. Stat. Phys. 115:1389-1421 (2004).
45. P. L. Garrido, S. Goldstein, and J. L. Lebowitz, Boltzmann entropy for dense fluids not in local equilibrium. Phys. Rev. 92:050602 (2004).
46. S. Goldstein and J. L. Lebowitz, On the (Boltzmann) entropy of nonequilibrium systems. Physica D 193:53-66 (2004).
47. G. M. Zaslavskil and R. Z. Sagdeev, Limits of statistical description of a nonlinear wave field. Soviet Physics JETP 25:718-724 (1967).
48. A. C. Newell, S. Nazarenko, and L. Biven, Wave turbulence and intermittency. Physica D 152-153:520-550 (2001).
49. Y. Choi, Y. V. Lvov, and S. Nazarenko, Joint statistics of amplitudes and phases in wave turbulence. Physica D 201:121-149 (2005).
50. Y. Choi, Y. V. Lvov, S. Nazarenko, and B. Pokorni, Anomalous probability of large amplitudes in wave turbulence, preprint, arXiv:math-ph/0404022.
51. Y. Choi, Y. V. Lvov, and S. Nazarenko, Wave turbulence, preprint, arXiv:math-ph/0412045.
52. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Dynamical Systems III, Springer, Berlin 1988.