Relaxation of weakly collisional plasma: continuous spectra, discrete eigenmodes, and the decay of echoes

Abstract: The relaxation of a weakly collisional plasma, which is of fundamental interest to laboratory and astrophysical plasmas, can be described by the Boltzmann-Poisson equations with the Lenard-Bernstein collision operator. We perform a perturbative analysis of these equations, and obtain exact analytic solutions that resolve long-standing controversies regarding the impact of weak collisions on the continuous spectra, discrete Landau eigenmodes, and the decay of plasma echoes. We retain both damping and diffusion terms in the collision operator. We find that the linear response is a temporal convolution of a continuum that depends on the continuous velocities of particles (crucial for the plasma echo) and discrete modes that describe coherent oscillations of the entire system. The discrete modes are exponentially damped over time due to collective effects or wave-particle interactions (Landau damping) as well as collisional dissipation. The continuum is also damped by collisions, but somewhat differently. Up to a collision time, the inverse of the collision frequency $\nu_{\mathrm{c}}$, the continuum decay is driven by the diffusion of particle velocities and occurs cubic exponentially over a timescale $\sim \nu^{-1/3}_{\mathrm{c}}$. After a collision time, however, the continuum decay is driven by the damping of velocities and diffusion of positions, and occurs exponentially over a timescale $\sim \nu_{\mathrm{c}}$. This slow exponential decay damps perturbations the most on scales $\lambda$ comparable to the mean free path $\lambda_{\mathrm{c}}$, but very slowly on larger scales. This establishes local thermal equilibrium, the essence of the fluid limit, and enhances the detectability of the plasma echo. The long-term decay is driven by the discrete modes for $\lambda<\lambda_{\mathrm{c}}$, but, by the slowly decaying continuum and the least damped mode for $\lambda>\lambda_{\mathrm{c}}$.