Probability density function model equation for particle charging in a homogeneous dusty plasma.

In this paper, we use the direct interaction approximation (DIA) to obtain an approximate integrodifferential equation for the probability density function (PDF) of charge (q) on dust particles in homogeneous dusty plasma. The DIA is used to solve the closure problem which appears in the PDF equation due to the interactions between the phase space density of plasma particles and the phase space density of dust particles. The equation simplifies to a differential form under the condition when the fluctuations in phase space density for plasma particles change very rapidly in time and is correlated for very short times. The result is a Fokker-Planck type equation with extra terms having third and fourth order differentials in q, which account for the discrete nature of distribution of plasma particles and the interaction between fluctuations. Approximate macroscopic equations for the time evolution of the average charge and the higher order moments of the fluctuations in charge on the dust particles are obtained from the differential PDF equation. These equations are computed, in the case of a Maxwellian plasma, to show the effect of density fluctuations of plasma particles on the statistics of dust charge.