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

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 ene...

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Bibliographic Details
Published inCommunications in mathematical physics Vol. 276; no. 1; pp. 23 - 49
Main Authors Castella, F., Degond, P., Goudon, Th
Format Journal Article
LanguageEnglish
Published Heidelberg Springer 01.11.2007
Springer Verlag
Subjects
Analysis of PDEs
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematical analysis
Mathematical methods in physics
Mathematics
Other topics in mathematical methods in physics
Partial differential equations
Physics
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Kinetic equations
Hamiltonians
Asymptotic behavior
Homogenization
Diffusion process
Evolution equation
Diffusion equation
Asymptotic solution
Mathematical physics
RANDOM-MEDIA
TRANSPORT-EQUATIONS
KINETIC-EQUATIONS
APPROXIMATION
QUANTUM BOLTZMANN-EQUATION
CONVERGENCE
VON-NEUMANN-EQUATION
DETERMINISTIC FRAMEWORK
BLOCH MODEL
HOMOGENIZATION
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ISSN0010-3616
1432-0916
DOI10.1007/s00220-007-0339-7

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Summary: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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-007-0339-7