Effective distribution functions for electrostatic waves in dusty plasmas with a dust-size distribution

The kinetic theory for the electrostatic modes of dusty plasmas with a distribution of grain sizes is examined. It is assumed that the size distribution predominantly decreases exponentially with the mass for large sizes, and that a power law prevails for small sizes. Thermodynamic equilibrium leads to Maxwellian distributions over velocity with a fixed temperature and continuously varying mass. Smaller particles have higher thermal velocity and dominate the tail of the velocity distribution. The contribution of the dust component to the dispersion function is found to be "non-Maxwellian" and is equivalent to that for a kappa (generalized Lorentzian) distribution of monosized particles. Known results fur kappa distributions may be exploited. However, the nonlinear response of the charge density of the dust to an electrostatic potential is quite different to that of a monosized kappa distribution. In general, the definition of an effective dust distribution function for linearized electrostatic modes leads to a useful straightforward procedure to find the dispersion function. It is important to realize that the combined effects of velocity and size distribution can, in general, strongly modify the kinetic behavior of the plasma dust component.