Diffusion dynamics of classical systems driven by an oscillatory force

Journal of Statistical Physics

Volumn 124, Issue 2-4, 2006, Pages 913-950

  Castella, F ^a   Degond, P ^b   Goudon, Th ^c  

^a UNIVERSITÉ DE RENNES 1   (France)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^c INRIA   (France)

Author keywords

Diffusion limit; Homogenization; Kinetic equation

Indexed keywords

EID: 33751059243     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-006-9071-5     Document Type: Article

References (41)

1. R. Alexandre, Some results in homogenization tackling memory effects, Asymptot. Anal. 15(3-4):229-259 (1997).
2. R. Alexandre, Asymptotic behaviour of transport equations, Appl. Anal. 70(3-4):405-430 (1999).
3. G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23:1482-1518 (1992).
4. Y. Amirat, K. Hamdache, and A. Ziani, On homogenization of ordinary differential equations and linear transport equations. In Calculus of Variations, Homogenization and Continuum Mechanics (Marseille, 1993) Sci: Adv. Math. Appl. Sci., vol. 18, (World Sci. Publishing, 1994) pp. 29-50.
5. G. Bal, G. Papanicolaou, and L. Ryzhik, Radiative transport limit for the random Schrödinger equation, Nonlinearity 15(2):513-529 (2002).
6. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures. Studies in Mathematics and its Applications, Vol. 5 (North-Holland, 1978).
7. C. Bardos, L. Dumas, and F. Golse, Diffusion approximation for billiards with totally accommodating scatterers, J. Statist. Phys. 86(1-2):351-375 (1997).
8. B. Bidégaray, F. Castella, and P. Degond, From Bloch model to the rate equations, Discrete Contin. Dyn.Syst. 11(1):1-26 (2004).
9. B. Bidégaray, F. Castella, E. Dumas, and M. Gisclon, From Bloch model to the rate equations II: the case of almost degenerate energy levels, Math. Models Methods Appl. Sci. 14(12):1785-1817 (2004).
10. L. A. Bunimovich, N. I. Chernov, and Ya. G. Sinai, Statistical properties of two-dimensional hyperbolic billiards, Uspekhi Mat. Nauk 46(4):43-92, (1991);
11. translation in Russian Math. Surveys 46(4):47-106(1991).
12. J. Casado-Diaz and I. Gayte, The two-scale convergence method applied to generalized Besicovitch spaces, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458(2028):2925-2946 (2002).
13. F. Castella, On the derivation of a Quantum Boltzmann Equation from the periodic von Neumann equation. Mod. Math. An. Num. 33(2):329-349 (1999).
14. F. Castella, From the von Neumann equation to the Quantum Boltzmann equation in a deterministic framework, J. Stat. Phys. 104(1/2):387-447 (2001).
15. F. Castella and P. Degond Convergence de l'équation de vonNeumann vers l'équationde Boltzmann Quantique dans un cadre déterministe, C. R. Acad. Sci., t. 329, sér. I, 231-236 (1999).
16. N. Dunford and J. Schwartz, Linear Operators (John Wiley & Sons, 1988).
17. L. Erdös and H. T. Yau, Linear Boltzmann equation as scaling limit of quantum Lorentz gas, Advances in Differential Equations and Mathematical Physics, Contemporary Mathematics 217:135-155 (1998).
18. L. Erdös and H. T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Commun. Pure Appl. Math. 53:667-735 (2000).
19. L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A 111:359-375 (1989).
20. L. C. Evans, Periodic homogenization of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 120:245-265 (1992).
21. E. Frenod and K. Hamdache, Homogenisation of transport kinetic equations with oscillating potentials, Proc. Roy. Soc. Edinburgh Sect. A 126(6):1247-1275 (1996).
22. T. Goudon and F. Poupaud, Approximation by homogeneization and diffusion of kinetic equations, Comm. P.D.E. 26:537-570 (2001).
23. T. Goudon and F. Poupaud, Homogenization of transport equations: weak mean field approximation, SIAM J. Math Anal. 36:856-881 (2004).
24. T. Goudon and F. Poupaud, Homogenization of transport equations: A simple PDE approach to the Kubo formula, Preprint.
25. .T. B. Keller, G. Papanicolaou, and L. Ryzhik, Transport equations for elastic and other waves in random media, Wave Motion 24(4):327-370 (1996).
26. O. Lanford III, The evolution of large classical systems, in: Moser, J. (ed) Dynamical Systems, Theory and Applications. Lectures Notes in Physics 35 (Springer, 1975) pp. 1-111.
27. G. Loeper and A. Vasseur, Electric turbulence in a plasma subject to a strong magnetic field, Asymptot. Anal. 40(1):51-65 (2004).
28. R. Loudon, The Quantum Theory of Light (Oxford Science Publications, 1983).
29. J. W. Milnor, Topology front the Differentiate Viewpoint (The University Press of Virginia, 1963).
30. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20:608-623 (1989).
31. G. Nguetseng, Almost periodic homogenization: asymptotic analysis of a second order elliptic equation, Publications Mathématiques du laboratoire d'analyse numérique, Université Yaoundé I.
32. B. Perthame and L. Ryzhik, The quantum scattering limit for a regularized Wigner equation, Preprint.
33. M. Petrini, Homogenization of a linear transport equation with time depending coefficient, Rend. Sem. Mat. Univ. Padova 101:191-207 (1999).
34. F. Poupaud and A. Vasseur, Classical and quantum transport in random media, Journal de Mathématiques Pures et Appliquées 82(6):711-748 (2003).
35. E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127 (Springer-Verlag, 1980).
36. J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Appl. Math. Sci. 59 (Springer-Verlag, 1985).
37. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, 1991).
38. L. Tartar, H-convergence et compacité par compensation, Cours Peccot, Collège de France, March 1977.
39. Partially written in: F. Murat, H-convergence, Séminaire d'Analyse Fonctionelle et Numérique 1977-78, Université d'Alger, multicopied 34 p.
40. English translation in F. Murat, L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, A. Cherkaev - R. V Kohn eds., Progress in Nonlinear Differential Equations Appl., Vol. 31 (Birkhäuser, 1997) pp. 21-43.
41. L. Tartar, Remarks on homogenization in Homogenization and Effective Moduli of Material and Media, IMA Vol. in Math, and Appl. (Springer, 1986) pp. 228-246.