Green's Function for the Linearized One-Dimensional Krook Equation with Electric Forces

An integral representation is obtained for the Green's function for the linearized one-dimensional Krook equation with the induced electric field of the medium included. Various asymptotic expansions in time are then obtained. When the plasma frequency is set to zero, slightly modified hydrodynamic modes appear. For nonzero plasma frequency, only plasma oscillations unaffected by the collisions are present. Finally, the initial value problem corresponding to an initial wave packet of approximate wavenumber k is considered. For times long, but not too long, plasma oscillations are present for which the frequency and wavenumber satisfy the usual Landau dispersion relation for small wavenumber. After a sufficiently long time, the solution behaves like the Green's function itself and exhibits Landau damping.