Not to normal order-notes on the kinetic limit for weakly interacting quantum fluids

Journal of Statistical Physics

Volumn 134, Issue 5-6, 2009, Pages 1133-1172

  Lukkarinen, Jani ^a   Spohn, Herbert ^b  

^a UNIVERSITY OF HELSINKI   (Finland)

^b TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Boltzmann nordheim equation; Kinetic theory; Quantum bbgky hierarchy; Time dependent perturbation theory; Uehling uhlenbeck equation; Weakly interacting bosons; Weakly interacting fermions

Indexed keywords

EID: 67349146432     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-009-9682-8     Document Type: Article

References (40)

1. Nordheim, L.W.: On the kinetic method in the new statistics and its application in the electron theory of conductivity. Proc. R. Soc. 119, 689-698 (1928)
2. Peierls, R.E.: Zur kinetischen Theorie der Wärmeleitung in Kristallen. Ann. Phys. 3, 1055-1101 (1929)
3. Uehling, E.A., Uhlenbeck, G.E.: Transport phenomena in Einstein-Bose and Fermi-Dirac gases. Phys. Rev. 43, 552-561 (1933)
4. Landau, L.D.: The transport equation in the case of Coulomb interactions. Phys. Z. Sowj. Union 10, 154 (1936). (As no. 24 in: Collected Papers of L.D. Landau, edited by D. TerHaar, Pergamon Press, Oxford, 1965)
5. van Hove, L.: Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21,517-540(1955)
6. Prigogine, I.: Nonequilibrium Statistical Mechanics. Wiley, New York (1962)
7. Hugenholtz, N.M.: Derivation of the Boltzmann equation for a Fermi gas. J. Stat. Phys. 32, 231-254 (1983)
8. Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Lecture Notes in Physics, vol. 38, pp. 1-111. Springer, Berlin (1975)
9. Davies, E.B.: Quantum Theory of Open Systems. Academic Press, London (1976)
10. Hilbert, D.: Mathematische Probleme. In: Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. Vandenhoeck & Ruprecht, S. 253-297 (1900)
11. Robert, D.: Semi-classical approximation in quantum mechanics. A survey of old and recent mathematical results. Helv. Phys. Acta 71, 44-116 (1998)
12. Fröhlich, J., Graffi, S., Schwarz, S.: Mean-field and classical limit of many body Schrödinger dynamics for bosons. Commun. Math. Phys. 271, 681-697 (2007)
13. Fröhlich, J., Knowles, A., Pizzo, A.: Atomism and quantization. J. Phys. A 40, 3033-3045 (2007)
14. Erdó́s, L., Schlein, B.: Quantum dynamics with mean field interaction: nonlinear Hartree equation. J. Stat. Phys. (2009, to appear)
15. Lukkarinen, J., Spohn, H.: Weakly nonlinear Schrödinger equation with random initial data. arXiv:0901.3283 (2009)
16. Streater, R.F.: On certain nonrelativistic quantized fields. Commun. Math. Phys. 7, 93-98 (1968)
17. Ginibre, J.: Some applications of functional integration in statistical mechanics. In: DeWitt, C., Stova, R. (eds.) Statistical Mechanics and Quantum Field Theory, pp. 327-428. Gordon & Breach, New York (1971)
18. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, 2nd edn. Text and Monographs in Physics, vol. 2. Springer, Berlin (1997)
19. Ho, T.G., Landau, L.J.: Fermi gas on a lattice in the van Hove limit. J. Stat. Phys. 87, 821-845 (1997)
20. Erdó́s, L., Salmhofer, M., Yau, H.T.: On the quantum Boltzmann equation. J. Stat. Phys. 116, 367-380 (2004)
21. Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124, 1041-1104 (2006)
22. Botvich, D.D., Malishev, V.A.: Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi gas. Commun. Math. Phys. 91, 301-312 (1983)
23. Fröhlich, J., Merkli, M., Ueltschi, D.: Dissipative transport: thermal contacts and tunnelling junctions. Ann. Henri Poincaré 4, 897-945 (2003)
24. Erdó́s, L., Yau, H.T.: Linear Boltzmann equation as the weak coupling limit of a random Schröldinger equation. Commun. Pure Appl. Math. 53, 667-735 (2000)
25. Benedetto, D., Castella, F., Esposito, R., Pulvirenti, M.: From the N-body Schrodinger equation to the quantum Boltzmann equation: a term-by-term convergence result in the weak coupling regime. Commun. Math. Phys. 277, 1-44 (2008)
26. Benedetto, D., Castella, F., Esposito, R., Pulvirenti, M.: Some considerations on the derivation of the nonlinear quantum Boltzmann equation. J. Stat. Phys. 116, 381-410 (2004)
27. Benedetto, D., Castella, F., Esposito, R., Pulvirenti, M.: On the weak coupling limit for bosons and fermions. Math. Mod. Meth. Appl. Sci. 15, 1811-1843 (2005)
28. Zakharov, V.E., L'vov, V.S., Falkovich, G.: Kolmogorov Spectra of Turbulence: I. Wave Turbulence. Springer, Berlin (1992)
29. Salmhofer, M.: Clustering of fermionic truncated expectation values via functional integration. arXiv:0809.3517 (2008)
30. Chen, T.: Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3. J. Stat. Phys. 120, 279-337 (2005)
31. Lukkarinen, J.: Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations. J. Math. Pures Appl. 87, 193-225 (2007)
32. Erdó́s, L., Salmhofer, M.: Decay of the Fourier transform of surfaces with vanishing curvature. Math. Z. 257, 261-294 (2007)
33. Spohn, H.: Collisional invariants for the phonon Boltzmann equation. J. Stat. Phys. 124, 1131-1135 (2006)
34. Escobedo, M., Mischler, S., Valle, M.: Homogeneous Boltzmann equation in quantum relativistic kinetic theory. Electron J. Differ. Equ. 4, 1-85 (2003)
35. Dolbeaut, J.: Kinetic models and quantum effects: a modifed Boltzmann equation for Fermi-Dirac particles. Arch. Ration. Mech. Anal. 127, 101-131 (1994)
36. Josserand, C., Pomeau, Y., Rica, S.: Self-similar singularities in the kinetics of condensation. J. Low Temp. Phys. 145, 231-265 (2006)
37. Semikoz, D.V., Tkachev, I.I.: Condensation of bosons in the kinetic regime. Phys. Rev. D 55, 489-502 (1997)
38. Spohn, H.: Kinetics of the Bose-Einstein condensation. arXiv:0809.4551 (2008)
39. Lu, X.G.: On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles. J. Stat. Phys. 116, 1597-1649 (2004)
40. Lu, X.G.: The Boltzmann equation for Bose-Einstein particles: velocity concentration and convergence to equilibrium. J. Stat. Phys. 119, 1027-1067 (2005)