Numerical investigation of Landau damping in dynamical Lorentz gases

Abstract

We investigate numerically the behavior of a particle, or a set of particles, which exchange momentum and energy with the environment, described as a transverse vibrational field. The large time behavior is characterized, in some specific circumstances, by means of effective friction force and Landau damping. In order to discuss these issues on numerical grounds, we set up a dedicated numerical method. The scheme couples a Finite Element Method for the wave equation, with an appropriate transparent boundary condition that preserves the dispersive effects driving the asymptotic behavior of the system, and a symplectic scheme or a Semi-Lagrangian method for the evolution of the particles. The time discretisation is constructed to capture accurately the energy exchanges. The numerical simulations shed new light on the theoretical results and bring out clearly the role of the parameters of the models.

On quantum dissipative systems: ground states and orbital stability

“…and L 2 -super-critical in the case d ⩾ 3. Hence, this formal limit suggests different behaviors for the Hartree equation (when a smooth potential is considered), depending on the dimension d. Even if the continuity with respect to Σ as Σ → δ 0 is certainly wrong when d ⩾ 2 -( 17) admits solutions which blow up in finite time while solutions of ( 13) are globally defined when Σ is smooth -our analysis shows several differences between the case d = 1 and d ⩾ 2, which can be understood from the formal asymptotic to (17). It is thus not surprising that our main results, Theorem 2.…”

“…Quite surprisingly -mind the sign κ > 0 -this corresponds to an attractive dynamics. This unexpected connection guides the intuition to establish further features of the solutions of the Vlasov-wave system; it particular, they exhibit Landau damping phenomena [17,18]. The analysis of these models, either for a single particle or the kinetic description, brings out the critical role of the wave speed c > 0 and the dimension n of the space for the wave equation.…”

“…Hence it can be used as a reference state to understand the dynamics of (1a)-(1b), which is expected to induce quite intricate deformations of the solitary wave. This issue is further discussed on numerical grounds in [17].…”

Long time behaviour of interacting particles through a vibrating medium: comparison between the N-particle systems and the natural kinetic equation dynamics

“…The proof also allows us to state the following points: (18). • If one remove the hypothesis σ 2 ≥ 0 from (A1), the computations leading to (18) also make sens for n = 1, 2 when R d σ 2 dx = 0.…”

“…The orbital stability of a family of equilibrium states has been investigated in [9]. More recently the Landau damping has been proved in [18] and some numerical simulation are in preparation in [19]. The relaxation to equilibrium with an additional dissipative Fokker-Planck operator has been established in [2] thanks to hypocoercive methods.…”