Landau damping and energy conservation

Landau damping is a collisionless dissipation process for electrostatic plasma waves governed by the linearized Vlasov-Poisson equations. Its physical mechanism is still being debated. It is commonly believed that Landau damping is caused by the energy exchange between waves and particles. However, this picture is not compatible with the linearized Vlasov-Poisson system, because the kinetic energy of particles in the linearized Vlasov-Poisson system is conserved, and the energy of the electrical field cannot be converted into the kinetic energy for the particles. We derived an energy relation that is fully consistent with the linearized Vlasov-Poisson system. It is shown that this energy relation is an exact statement of Landau damping, and thus the most accurate physical picture for it.