A Parametrically Driven Particle in the Presence of a Stochastic Source:
A Model for Thermal Equilibrium in a Paul Trap
A. Izmailov, S. Arnold, and A.S. Myerson
Phys.Rev.E 50, 702-708(1994)

An analytical approach is developed to consider confined motion of a charged microparticle within the Paul trap (an electrodynamic levitator trap) in an atmosphere near the standard temperature and pressure. The suggested approach is based on a second-order linear stochastic differential equation which describes damped microparticle motion subjected to the combined periodic parametric and random external excitations. To solve this equation a new ansatz is developed. This ansatz is a generalization of the Bogoliubov-Krylov decomposition technique, which is usually used to reduce th order of a differential equation. The solution is obtained in the long term imaging limit by applying the Bogoliubov general averaging principle. In spite of the second-order form of the initial stochastic differential equatioon, the microparticle motion can be understood as a one dimensional Markov process. Comparison in the long time imaging limit of the calculated data obtained from the analytically derived expression for the standard deviation of the confined microparticle stochastic motion with the experimentally obtained data demonstrates asymptotic agreement for regions where the dimensionless parameter k is much less than 1 (kb 0.005). Simple extremum analysis of the expression obtained for the standard deviation reveals that for the particular case of a large drag parameter a (a>>8[12] SIZE=1>1/2) there is a minimum in the standard deviation which is only a dependent.