Landau contours and collisionless damping

Use exp(ikx − iωt) with positive real k. A one-sided time transform of an initial perturbation is first defined sufficiently high in the upper complex-frequency half-plane for convergence. There the pole v = ω/k lies above the real velocity axis, so the ordinary real-axis integral is nonsingular. The causal response at other frequencies is obtained by analytic continuation from that initial domain, rather than by defining a fresh integral independently at each root.

As the pole moves downward, deform the velocity contour to keep passing below it. At a real phase velocity approached from above, the integral equals its principal value plus an imaginary pole term. The sign depends on the denominator and Fourier convention; reversing only one convention is an error. Smoothness or analyticity conditions on equilibrium and initial data determine how far continuation and pole-dominated asymptotics are justified. A damped Landau pole is not automatically a normal mode of the original real-velocity operator.

Write the complex frequency as a real part plus i times gamma, so positive gamma means growth. If the imaginary dielectric part and gamma are small, expand the dispersion equation around a simple real root of its real part. The damping rate is minus the imaginary dielectric value divided by the real frequency derivative. This expansion requires a nonzero derivative and a separation between the oscillation and growth or decay timescales.

For the positive-frequency Langmuir branch of a non-drifting Maxwellian, the real dielectric derivative is positive. The Maxwellian slope at positive phase velocity is negative, which gives a positive imaginary dielectric part under the stated convention and therefore negative gamma. A positive resonant slope can supply energy to this positive-energy branch instead. That slope rule is not universal for every wave or equilibrium: the wave-energy sign and other resonant species also enter the denominator and total response.

Different velocities accumulate different spatial Fourier phases during streaming. Their contributions can cancel in the velocity integral even when none has individually decayed. The density moment and its electric field may therefore become small while the distribution retains oscillatory velocity structure. An exactly solvable free-streaming Gaussian perturbation demonstrates this cancellation, but it must be distinguished from self-consistent Landau dynamics: the latter also includes electric forces and collective response.

Ideal Vlasov evolution preserves fine-grained entropy and can support echoes when appropriately prepared perturbations rephase. Weak collisions act efficiently on the fine velocity scales produced by mixing and can then produce irreversible entropy. A finite-grid calculation can imitate this smoothing unintentionally. Exponential field decay over a fitted interval should thus be checked against velocity resolution, initial perturbation smoothness, collisions, and the nonlinear trapping time. Linear collisionless damping is a precise asymptotic statement with assumptions, not a claim that all disturbances vanish forever.