Large Time Dynamics of a Classical System Subject to a Fast Varying Force - Castella, F.; Degond, P.; Goudon, Th (Communications in mathematical physics, 2007)
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.