Stochastic heating of dust particles in complex plasmas as an energetic instability of a harmonic oscillator with random frequency

The paper describes the occurrence of stochastic heating of dust particles in dusty plasmas as an energy instability due to the correlations between dust grain charge and electric field fluctuations. The possibility that the mean energy ("temperature") of dust particles can grow in time, has been found both from the self-consistent kinetic description of dusty plasmas taking into account charge fluctuations~[de Angelis U, Ivlev A V, Tsytovich V N and Morfill G E 2005 {\it Phys. Plasmas} {\bf 12}(5) 052301] and from a Fokker-Planck approach to systems with variable charge~[Ivlev A V, Zhdanov S K, Klumov B A and Morfill G E 2005 {\it Phys. Plasmas} {\bf 12}(9) 092104]. Here a different derivation is given by using the mathematical techniques of the so called multiplicative stochastic differential equations. Both cases of "fast" and "slow" fluctuations are discussed.