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Communications in Mathematical Physics

, Volume 276, Issue 1, pp 23–49 | Cite as

Large Time Dynamics of a Classical System Subject to a Fast Varying Force

  • F. CastellaEmail author
  • P. Degond
  • Th. Goudon
Article

Abstract

We investigate the asymptotic behavior of solutions to a kinetic equation describing the evolution of particles subject to the sum of a fixed, confining, Hamiltonian, and a small, time-oscillating, perturbation. The equation also involves an interaction operator which acts as a relaxation in the energy variable. This paper aims at providing a classical counterpart to the derivation of rate equations from the atomic Bloch equations. In the present classical setting, the homogenization procedure leads to a diffusion equation in the energy variable, rather than a rate equation, and the presence of the relaxation operator regularizes the limit process, leading to finite diffusion coefficients. The key assumption is that the time-oscillatory perturbation should have well-defined long time averages: our procedure includes general “ergodic” behaviors, amongst which periodic, or quasi-periodic potentials only are a particular case.

Keywords

Compactness Property Stochastic Average Homogenization Procedure Relaxation Term Linear Boltzmann Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Al.
    Alexandre R. (1999). Asymptotic behaviour of transport equations. Appl. Anal. 70(3–4): 405–430 zbMATHCrossRefMathSciNetGoogle Scholar
  2. A.
    Allaire G. (1992). Homogenization and two-scale convergence. SIAM J. Math. Anal. 23: 1482–1518 zbMATHCrossRefMathSciNetGoogle Scholar
  3. BPR.
    Bal G., Papanicolaou G. and Ryzhik L. (2002). Radiative transport limit for the random Schrödinger equation. Nonlinearity 15(2): 513–529 zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. BDG.
    Bardos C., Dumas L. and Golse F. (1997). Diffusion approximation for billiards with totally accommodating scatterers. J. Stat. Phys. 86(1–2): 351–375 zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. BLP.
    Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures. Studies in Mathematics and its Applications, 5, Amsterdam: North-Holland, 1978Google Scholar
  6. BCD.
    Bidégaray B., Castella F. and Degond P. (2004). From Bloch model to the rate equations. Discrete Contin. Dyn. Syst. 11(1): 1–26 zbMATHMathSciNetCrossRefGoogle Scholar
  7. BCDG.
    Bidégaray B., Castella F., Dumas E. and Gisclon M. (2004). From Bloch model to the rate equations II: the case of almost degenerate energy levels. Math. Models Methods Appl. Sci. 14(12): 1785–1817 zbMATHCrossRefMathSciNetGoogle Scholar
  8. BSC.
    Bunimovich, L.A., Chernov, N.I., Sinai, Ya.G.: Statistical properties of two-dimensional hyperbolic billiards. Usp. Mat. Nauk 46, no. 4, 43–92, (1991); translation in Russ. Math. Surv. 46(4), 47–106 (1991)Google Scholar
  9. Ca1.
    Castella F. (2001). From the von Neumann equation to the Quantum Boltzmann equation in a deterministic framework. J. Stat. Phys. 104(1/2): 387–447 zbMATHCrossRefMathSciNetGoogle Scholar
  10. Ca2.
    Castella F. (2002). From the von Neumann equation to the Quantum Boltzmann equation II: identifying the Born series. J. Stat. Phys. 106(5/6): 1197–1220 zbMATHCrossRefMathSciNetGoogle Scholar
  11. CD.
    Castella, F., Degond, P.: Convergence de l’équation de von Neumann vers l’équation de Boltzmann Quantique dans un cadre déterministe. C. R. Acad. Sci., t. 329, sér. I, 231–236 (1999)Google Scholar
  12. CDG.
    Castella F., Degond P. and Goudon T. (2006). Diffusion dynamics of classical systems driven by an oscillatory force. J. Stat. Phys. 124(2–4): 913–950 zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. CIP.
    Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences, 106, New York: Springer-Verlag, 1994Google Scholar
  14. EY1.
    Erdös, L., Yau, H.T.: Linear Boltzmann equation as scaling limit of quantum Lorentz gas, Advances in Differential Equations and Mathematical Physics, Contemporary Mathematics 217, Providence, RI: Amer. Math. Soc., pp.135–155 (1998)Google Scholar
  15. EY2.
    Erdös L. and Yau H.T. (2000). Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Commun. Pure Appl. Math. 53: 667–735 zbMATHCrossRefGoogle Scholar
  16. FH.
    Frenod E. and Hamdache K. (1996). Homogenisation of transport kinetic equations with oscillating potentials. Proc. Roy. Soc. Edinburgh Sect. A 126(6): 1247–1275 zbMATHMathSciNetGoogle Scholar
  17. GP1.
    Goudon T. and Poupaud F. (2001). Approximation by homogeneization and diffusion of kinetic equations. Comm. P.D.E. 26: 537–570 zbMATHCrossRefMathSciNetGoogle Scholar
  18. GP2.
    Goudon T. and Poupaud F. (2004). Homogenization of transport equations: weak mean field approximation. SIAM J. Math Anal. 36: 856–881 zbMATHCrossRefMathSciNetGoogle Scholar
  19. GP3.
    Goudon T. and Poupaud F. (2007). Homogenization of transport equations: a simple PDE approach to the Kubo formula. Bull.Sci. Math. 131: 72–88 zbMATHCrossRefMathSciNetGoogle Scholar
  20. KPR.
    Keller J.B., Papanicolaou G. and Ryzhik L. (1996). Transport equations for elastic and other waves in random media. Wave Motion 24(4): 327–370 zbMATHCrossRefMathSciNetGoogle Scholar
  21. Li.
    Lions, P.-L.: Mathematical topics in fluid mechanics, Vol. 2, Oxford: Oxford univ. Press, Oxford Science Publications 19, 1996–98Google Scholar
  22. LV.
    Loeper G. and Vasseur A. (2004). Electric turbulence in a plasma subject to a strong magnetic field. Asymptot. Anal., 40(1): 51–65 zbMATHMathSciNetGoogle Scholar
  23. Lo.
    Loudon, R.: The quantum theory of light. Oxford Science Publications, Oxford: Oxford univ. Press, 1983Google Scholar
  24. Mi.
    Milnor, J.W.: Topology from the differentiable viewpoint. The University Press of Virginia, 1963Google Scholar
  25. Ng.
    Nguetseng G. (1989). A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20: 608–623 zbMATHCrossRefMathSciNetGoogle Scholar
  26. PR.
    Perthame B. and Ryzhik L. (2004). The quantum scattering limit for a regularized Wigner equation. Math. Appl. Anal. 11: 447–464 zbMATHMathSciNetGoogle Scholar
  27. PV.
    Poupaud F. and Vasseur A. (2003). Classical and quantum transport in random media. J. de Math. Pures et Appl 82(6): 711–748 zbMATHMathSciNetGoogle Scholar
  28. SV.
    Sanders, J.A., Verhulst, F.: Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences, Vol. 59, Berlin-Heidelberg-Newyork: Springer-Verlag, 1985Google Scholar
  29. Sp.
    Spohn H. (1991)Large scale dynamics of interacting particles, Berlin-Heidelberg-Newyork: SpringerzbMATHGoogle Scholar
  30. Ta.
    Taylor, G.I.: Diffusion by continuous movements, Proc. London Math. Soc., Ser., 2, 20, 196–211, 1923Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.IRISA & IRMARUniversité de Rennes 1Rennes CedexFrance
  2. 2.MIP, UMR 5640 (CNRS-UPS-INSA)Université Paul Sabatier - 118Toulouse CedexFrance
  3. 3.Team SIMPAF - INRIA Futurs & Labo. Paul Painlevé, UMR 8524Université des Sciences et Technologies Lille 1Villeneuve d’Ascq CedexFrance

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