A kinetic model of a dusty plasma with discrete charges

We study a kinetic model of a “dusty plasma” consisting of electrons and dust grains under the action of an external electrostatic field. An electron that hits a dust particle transfers its charge to the grain. Accordingly, the population of dust grains is subdivided in 0-charged (neutral), 1-charged, 2-charged, and so on. The model results into a nonlinear system of infinitely many kinetic evolution equations. The “concentrated parameter” approximation reduces it to a nonlinear system of infinitely many ODE's, whose well-posedness and asymptotic behavior are analyzed.