Abstract
For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, f, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad and, much refined, Cercignani argue for the existence of this limit on the basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic time span, the argument can be made mathematically precise following the seminal work of Lanford. In this article a corresponding program is undertaken for weakly nonlinear, both discrete and continuum, wave equations. Our working example is the harmonic lattice with a weakly nonquadratic on-site potential. We argue that the role of the Boltzmann f-function is taken over by the Wigner function, which is a very convenient device to filter the slow degrees of freedom. The Wigner function, so to speak, labels locally the covariances of dynamically almost stationary measures. One route to the phonon Boltzmann equation is a Gaussian decoupling, which is based on the fact that the purely harmonic dynamics has very good mixing properties. As a further approach the expansion in terms of Feynman diagrams is outlined. Both methods are extended to the quantized version of the weakly nonlinear wave equation.
The resulting phonon Boltzmann equation has been hardly studied on a rigorous level. As one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal Wigner functions. For three phonon processes such a result requires extra conditions on the dispersion law. We also outline the reasoning leading to Fourier’s law for heat conduction.
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References
R. E. Peierls, Zur kinetischen Theorie der Wärmeleitung in Kristallen. Annalen Physik 3:1055–1101 (1929).
G. Leibfried, Gittertheorie der mechanischen und thermischen Eigenschaften der Kristalle, Handbuch der Physik Band VII/1, Kristallphysik, ed. S. Flügge, Springer, Berlin 1955.
V. L. Gurevich, Transport in Phonon Systems, North-Holland 1986.
G. P. Srivastava, The Physics of Phonons, Adam Hilger, Bristol 1990.
J. Callaway, Quantum Theory of the Solid State, Academic Press 1974.
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).
C. J. Glassbrenner and G. Slack, Thermal conductivity of Silicon and Germanium from 3°K to the melting point. Phys. Rev. 134:A1058–A1069 (1964).
S. Fujita, Introduction to Nonequilibrium Quantum Statistical Mechanics, Saunders 1966.
L. Ryzhik, G. Papanicolaou, and J. B. Keller, Transport equations for elastic and other waves in random media. Wave Motion 24:327–370 (1996).
V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence: I Wave Turbulence. Springer, Berlin 1992.
P. Janssen, The Interaction of Waves and Wind, Cambridge University Press 2004.
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.
A. Mielke, Macroscopic behavior of microscopic oscillations in harmonic lattices, preprint 118 of SPP ’Analysis, Modeling and Simulation of Multiscale Problems’ (2004).
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).
T. Dudnikova and H. Spohn, Local stationarity for lattice dynamics in the harmonic approximation, preprint, arXiv:math-ph/0505031.
L. Erdös, M. Salmhofer, and H. T. Yau, On the quantum Boltzmann equation. J. Stat. Phys. 116:367–380 (2004).
O. Bratelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2, 2nd edition. Springer, Berlin 1996.
H. Spohn, unpublished notes, 2005.
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).
N. M. Hugenholtz, Derivation of the Boltzmann equation for a Fermi gas. J. Stat. Phys. 32:231–254 (1983).
N. T. Ho and L. J. Landau, Fermi gas on a lattice in the van Hove limit. J. Stat. Phys. 87:821–845 (1997).
C. Cercignani and G. M. Kremer, On relativistic collisional invariants. J. Stat. Phys. 96:439–445 (1999).
J. Lukkarinen and H. Spohn, Kinetic limit for wave propagation in a random medium, preprint, arXiv:math-ph/0505075.
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).
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).
T. Chen, Localization lengths and Boltzmann limit for the Anderson model at small disorder in dimension 3. J. Stat. Phys., online (2005).
T. Chen, Lr-Convergence of a random Schrödinger to a linear Boltzmann evolution, preprint, arXiv:math-ph/0407037.
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).
R. Brout, I. Prigogine, Statistical mechanics of irreversible processes part V: anharmonic forces. Physica 22:35–47 (1956).
I. Prigogine, Nonequilibrium Statistical Mechanics, Wiley-Interscience, New York 1962.
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.
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).
L. van Hove, Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21: 517–540 (1955).
E. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 49:749–759 (1932).
L. Erdös, M. Salmhofer, H.-T. Yau, Quantum diffusion of random Schrödinger evolution in the scaling limit, preprint, arXiv:math-ph/0502025.
G. Bal, T. Komorowski, and L. Ryzhik, Self-averaging of Wigner transforms in random media. Comm. Math. Phys. 242: 81–135 (2003).
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.
E. A. Jackson, Nonlinearity and irreversibility in lattice dynamics, Rocky Mount. J. Math. 8:127–196 (1978).
S. Lepri, R. Livi, and A. Politi, Thermal conductivity in classical low-dimensional lattices. Physics Reports 377:1–80 (2003).
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.
K. Aoki and D. Kusnezov, Nonequilibrium statistical mechanics of classical lattice phi4 field theory. Ann. Phys. 295:50–80 (2002).
G. R. Lee-Dadswell, B. G. Nickel, and C. G. Gray, Thermal conductivity and bulk viscosity in quartic oscillator chains, preprint (2005).
S. Lepri, Memory effects and heat transport in one-dimensional insulators. Eur. Phys. J. B18:441–446 (2000).
R. Lefevere and A. Schenkel, Perturbative analysis of anharmonic chains of oscillators out of equilibrium. J. Stat. Phys. 115:1389–1421 (2004).
P. L. Garrido, S. Goldstein, and J. L. Lebowitz, Boltzmann entropy for dense fluids not in local equilibrium. Phys. Rev. 92:050602 (2004).
S. Goldstein and J. L. Lebowitz, On the (Boltzmann) entropy of nonequilibrium systems. Physica D 193:53–66 (2004).
G. M. Zaslavskiui and R. Z. Sagdeev, Limits of statistical description of a nonlinear wave field. Soviet Physics JETP 25:718–724 (1967).
A. C. Newell, S. Nazarenko, and L. Biven, Wave turbulence and intermittency. Physica D 152–153:520–550 (2001).
Y. Choi, Y. V. Lvov, and S. Nazarenko, Joint statistics of amplitudes and phases in wave turbulence. Physica D 201:121–149 (2005).
Y. Choi, Y. V. Lvov, S. Nazarenko, and B. Pokorni, Anomalous probability of large amplitudes in wave turbulence, preprint, arXiv:math-ph/0404022.
Y. Choi, Y. V. Lvov, and S. Nazarenko, Wave turbulence, preprint, arXiv:math-ph/0412045.
V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Dynamical Systems III, Springer, Berlin 1988.
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Spohn, H. The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics. J Stat Phys 124, 1041–1104 (2006). https://doi.org/10.1007/s10955-005-8088-5
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DOI: https://doi.org/10.1007/s10955-005-8088-5