Substituting an integral over a smooth distribution function for the discrete sum over the particles in S ( \ mathbf { k }, \ omega ), yields the Vlasovian expression enabling the calculation of Landau damping ( section 6.4 of ).
12.
The two stream instability can be thought of as the inverse of Landau damping, where the existence of a greater number of particles that move slower than the wave phase velocity v _ { ph } as compared with those that move faster, leads to an energy transfer from the wave to the particles.
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