field introduces a fascinating complication into the motion of charged particles. And the particles trajectories become helices, spiraling around the magnetic lines of force. This severe alteration of the orbits tends to inhibit transport across the magnetic field. The mechanism of Landau damping does depend on the transfer of electric field energy to particles moving in phase with the wave. However, for cyclotron damping in electromagnetic plasmas, the electric field of the wave is perpendicular to the direction of the magnetic field and the particle drifts and accelerates the particle perpendicular to the drift direction.

In the following we recall Landau damping through gathering lots of physical literature and results of mathematical articles. The existence of a damping mechanism by which plasma particles absorb wave energy was found by L.D.Landau at the linear level, under the condition that the plasma is not cold and the velocity distribution is of finite extent. Next in linear case, many works from mathematical aspects found in \[9,15,25,28\] gave rigorous proofs under different assumptions. Later, a ground-breaking work for Landau damping was made by Mouhot and Villani in the nonlinear case. They gave the first and rigours proof of nonlinear Landau damping under the assumption 