Intrinsic Charge Fluctuations of Dust in Plasmas Containing Multiply Charged Ions

A master equation, formulated for states of the charge of a grain in a plasma containing various kinds of singly or multiply charged ions (Shotorban, Phys Plasmas, 2014) is presented. From the master equation, a Fokker-Planck equation is derived through van Kampen's system-size expansion method. The derived Fokker-Planck equation has a Gaussian solution with a mean and variance governed by two initial-value differential equations involving the ions and electron attachment rates. Also, a Langevin equation, statistically equivalent to the Fokker-Planck equation, and a discrete stochastic method, statistically representing the master equation, are developed to model the grain charge fluctuations in time. Grain charging in a plasma containing electrons, protons, and alpha particles is considered as a test problem. It is shown that the Gaussian solution is in very good agreement with the master equation's solution obtained numerically for this problem.