Fundamentals of Plasma Physics 2025, 43 - Echoes

Plasma wave echoes are a nonlinear effect related to Landau damping and entropy. Wave damping should destroy the information content of the wave by converting ordered motion to heat, increasing system entropy. At the same time, collisionless Vlasov equation conserves entropy, as collisions is the quantity underlying entropy. This is a contradiction at first glance. Entropy is the natural logarithm of the number of micro-states corresponding to a macro-state. Collisions cause the system to evolve through the micro-states and an observer of the macro-system sees no difference between the micro-state. To resolve the contradiction, the micro-states no longer maps to a single macro-state for Landau damping, as its system is only valid in one well-defined state. The macro-state would have to map to one micro-state, which means system evolution is discreet, and not continuous. As such, it doesn't convert ordered information to heat directly. Instead, macro-ordered information turns into micro-ordered information, rendering it invisible on the macro-scale, by scrambling the phase of the macro-info, encoding it into the micro-info, which is normally irreversible. The net velocity is generally written

This approach is closely related to Fourier transforms. Given appropriate initial conditions, Laplace transforms, which may be better suited, can be applied instead, creating instead

The ballistic terms superposition into typical destructive interference, quickly dropping to negligible terms at macro-scale.

The linearized density n₁(t) is defined analogously to the net velocity. Their product is nonlinear with a dependence on Re e^ikv₀t × Re e^{ik'v₀(t-τ)}. At time kt - k'(t - τ) = 0, the nonlinear ballistic term would be zero for all velocities. There, no phase mixing occurs. In the Vlasov-Poisson analysis of plasma waves, additional issues need to be considered. Equilibrium occurs at ∂_tf₀ + v∂_xf₀ - eE₀/m ∂_vf₀ = 0. Before expansion, we keep in mind that E₀, ∂_tf₀, ∂_xf₀ = 0. When periodic grids are inserted into a plasma, pulsed transiently to create a potential ϕ, the charged particles will move in response to this applied potential. The Poisson's has to react to it. ∇²ϕ₁ = -1/ε₀(grid charge density + plasma charge density).