Weak turbulence in dusty plasmas with collisional dust charging: Quasilinear wave-particle interaction.

Starting from the exact microscopic equations for an unmagnetized dusty plasma, where the dust charge is regarded as a new degree of freedom of the system, we present a self-consistent set of equations that is suitable for weak turbulence analyses, where we have considered that the dust is electrically charged by absorption of plasma particles through inelastic collisions. In the quasilinear limit of the wave-particle interaction, we find a modified equation for the spectral energy density of the plasma electric field, and two other equations of the spectral energy densities associated with the fluctuation of the dust distribution due to plasma particle absorption, and with the fluctuation of the dust charging current. Different dispersion relations are found respectively for each type of oscillation. The corresponding kinetic equations for the waves are given, as well as the temporal evolution equations of the dust and plasma particles.