Collisionless damping of a collective plasma perturbation by phase mixing and resonant exchange with particles near its phase velocity. In linear electrostatic response, the imaginary part of the dielectric function at real frequency is proportional to the derivative of the equilibrium phase-space distribution function at resonance. A velocity plateau removes this contribution on an interval; compact velocity support also permits undamped waves whose phase velocities lie outside that support.
For freely streaming particles, a spatial Fourier perturbation of a velocity distribution evolves with factors . Contributions from different velocities acquire different phases and can cancel in a density or field moment even while the fine-grained distribution retains structure. In a self-consistent plasma, resonant field–particle exchange gives Landau damping. This is distinct from collisional thermalization and from orbital phase mixing in a planetary ensemble.
A broad electron Maxwell-Boltzmann distribution of width plus a narrow beam of width centred at speed can have a positive resonant velocity slope. The beam slope is largest at . Treating the background slope as constant over the beam width gives the displayed density threshold. Replacing the Maxwellian tail at by its value at also requires ; width separation alone does not ensure that exponentially sensitive approximation. A positive slope is the local resonant-drive criterion, while an exact unstable mode also requires the kinetic dispersion relation.
A pole of the causally continued collective kinetic response, usually a zero of its scalar response denominator or response-matrix determinant. With time dependence , negative imaginary part gives damping, positive imaginary part growth, and zero imaginary part an undamped oscillation. Damped poles need not be regular discrete eigenmodes of the original collisionless operator: their contribution is obtained by continuing the initial-value response. The least damped excited pole can set the waiting time for a stationary forced response.
The velocity-integration contour implementing causal continuation of a kinetic response from the upper complex phase velocity half-plane. As a pole approaches the real axis, the contour passes below it. With denominator , the real-axis limit is . Continuing the pole below the axis keeps this causal prescription rather than simply replacing the contour by an undeformed real-axis integral.
