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.
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References
Alexandre R. (1999). Asymptotic behaviour of transport equations. Appl. Anal. 70(3–4): 405–430
Allaire G. (1992). Homogenization and two-scale convergence. SIAM J. Math. Anal. 23: 1482–1518
Bal G., Papanicolaou G. and Ryzhik L. (2002). Radiative transport limit for the random Schrödinger equation. Nonlinearity 15(2): 513–529
Bardos C., Dumas L. and Golse F. (1997). Diffusion approximation for billiards with totally accommodating scatterers. J. Stat. Phys. 86(1–2): 351–375
Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures. Studies in Mathematics and its Applications, 5, Amsterdam: North-Holland, 1978
Bidégaray B., Castella F. and Degond P. (2004). From Bloch model to the rate equations. Discrete Contin. Dyn. Syst. 11(1): 1–26
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
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)
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
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
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)
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
Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences, 106, New York: Springer-Verlag, 1994
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)
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
Frenod E. and Hamdache K. (1996). Homogenisation of transport kinetic equations with oscillating potentials. Proc. Roy. Soc. Edinburgh Sect. A 126(6): 1247–1275
Goudon T. and Poupaud F. (2001). Approximation by homogeneization and diffusion of kinetic equations. Comm. P.D.E. 26: 537–570
Goudon T. and Poupaud F. (2004). Homogenization of transport equations: weak mean field approximation. SIAM J. Math Anal. 36: 856–881
Goudon T. and Poupaud F. (2007). Homogenization of transport equations: a simple PDE approach to the Kubo formula. Bull.Sci. Math. 131: 72–88
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
Lions, P.-L.: Mathematical topics in fluid mechanics, Vol. 2, Oxford: Oxford univ. Press, Oxford Science Publications 19, 1996–98
Loeper G. and Vasseur A. (2004). Electric turbulence in a plasma subject to a strong magnetic field. Asymptot. Anal., 40(1): 51–65
Loudon, R.: The quantum theory of light. Oxford Science Publications, Oxford: Oxford univ. Press, 1983
Milnor, J.W.: Topology from the differentiable viewpoint. The University Press of Virginia, 1963
Nguetseng G. (1989). A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20: 608–623
Perthame B. and Ryzhik L. (2004). The quantum scattering limit for a regularized Wigner equation. Math. Appl. Anal. 11: 447–464
Poupaud F. and Vasseur A. (2003). Classical and quantum transport in random media. J. de Math. Pures et Appl 82(6): 711–748
Sanders, J.A., Verhulst, F.: Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences, Vol. 59, Berlin-Heidelberg-Newyork: Springer-Verlag, 1985
Spohn H. (1991)Large scale dynamics of interacting particles, Berlin-Heidelberg-Newyork: Springer
Taylor, G.I.: Diffusion by continuous movements, Proc. London Math. Soc., Ser., 2, 20, 196–211, 1923
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Communicated by J.L. Lebowitz
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Castella, F., Degond, P. & Goudon, T. Large Time Dynamics of a Classical System Subject to a Fast Varying Force. Commun. Math. Phys. 276, 23–49 (2007). https://doi.org/10.1007/s00220-007-0339-7
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DOI: https://doi.org/10.1007/s00220-007-0339-7