is that the number of particles that the wave velocity greatly exceeds their velocity is much larger than the number of particles whose velocity is slower than the wave velocity. The $(\mathbf{PSC})$ condition in above theorem is in consistent with the statement of the physics viewpoint. In fact, the $(\mathbf{PSC})$ condition and (0.11) imply that the number of resonant particles is exponentially small and their effect corresponding is weak. From the conclusion of Theorem (0.3), we know that, under this condition, most particles absorb energy from the wave, and then the damping occurs.

From the definition of the hybrid analytic norms and the proof of the following sections, there is still the phenomena of cyclotron damping for the nonlinear Vlasov-Maxwell equations, which is the same to that in the nonlinear Vlasov-Poisson equations but only in $\hat{z}$ direction, the position of the corresponding resonances translates 0 into $B_{0},$ and in the horizon direction, the action of the particles moves along circle that is the same to the linear case.

Linear Cyclotron Damping
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In this section, let us consider the following linear Vlasov equations in a uniform magnetic field, and recall the equations: $$\left\{\begin{array}{l}
\partial_{t}f+v\cdot\nabla_{x}f+\frac{q}{m}(v\times B_{0})\cdot\nabla_{v}f=-\frac{q}{m}(E+v\times B)\cdot\nabla_{v}f^{0},\\
\partial_{t}B=\nabla_{x}\times E,\quad \nabla\cdot B=0,\\
