ECHO, ECHo, Echo, echo... When echoes overwhelm Landau damping

The Liouville equation describing a collection of charged particles is time-reversible. In the weakly coupled limit, one can reduce this equation to a Fokker-Planck equation, which is irreversible. The problem of the fate of electromagnetic field fluctuations in a plasma in the limit of very weak irreversibility was addressed by Landau, who demonstrated that as long as there are some collisions (even if very rare), and in the absence of sources, gradients, etc, typical field fluctuations are damped with an easily calculated “collisionless” damping rate -- this is Landau damping. The energy of the field fluctuations is converted to particle energy; there is irreversible heating. Landau’s calculation is fine in the limit of small amplitude fluctuations, but what happens when the plasma is turbulent? I will show that in a typical nonlinear system (relevant to many physical observations), Landau damping is overwhelmed and ultimately arrested by turbulent “echoes”. This finding has important implications for detailed predictions of the heating (and in some cases, for the luminosity) of some interesting astrophysical plasmas.