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 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. 1. Setting of the Problem We consider the asymptotic behavior as ε goes to 0 of the solutions f (t, x, v) ≥ 0 to the following kinetic equation with relaxation term: ε2 ∂t f (t, x, v) + { H0(x, v), f ε } + ε { V ( t ε2 , x ) , f ε } = γ Q( f )(t, x, v), (1) where Q( f )(t, x, v) := P( f )(t, x, v)− f . (2) Here, the Poisson bracket {·, ·} stands as usual for { f, g} = ∇v f ·∇x g−∇x f ·∇vg. The position, resp. velocity, variables x , resp. v, both belong to the whole space Rd (d ≥ 1), and we shall often make use of the phase-space variable X = (x, v) ∈ R2d . Throughout this text, the Hamiltonian H0(X) ∈ C∞(R2d) is assumed given, and confining, i.e. lim |X |→∞ H0(X) = +∞. (3) 24 F. Castella, P. Degond, Th. Goudon The right-hand-side of (1)-(2) involves a projection operator P , whose value we define as P f (t, X) := [ f ε] (t, H0(X)), (4) where the quantity f (t, E) is the mean value of f ε over the energy shell SE := {X ∈ R2d s.t. H0(X) = E}, namely, f (t, E) := 1 h0(E) ∫ SE f (t, X) δ(H0(X)− E), (5)