  A linearised kinetic equation describing electrostatic perturbations of a
Maxwellian equilibrium in a weakly collisional plasma forced by a random source
is considered. The problem is treated as a kinetic analogue of the Langevin
equation and the corresponding fluctuation-dissipation theorem is derived. This
kinetic fluctuation-dissipation theorem reduces to the standard "fluid" one in
the regime where the Landau damping rate is small and the system has no real
frequency; in this case the simplest possible Landau-fluid closure of the
kinetic equation coincides with the standard Langevin equation. Phase mixing of
density fluctuations and emergence of fine scales in velocity space is
diagnosed as a constant flux of free energy in Hermite space; the
fluctuation-dissipation theorem for the perturbations of the distribution
function is derived, in the form of a universal expression for the Hermite
spectrum of the free energy. Finite-collisionality effects are included. This
work is aimed at establishing the simplest fluctuation-dissipation relations
for a kinetic plasma, clarifying the connection between Landau and
Hermite-space formalisms, and setting a benchmark case for a study of phase
mixing in turbulent plasmas.
