PARAMETRICALLY DRIVEN MICROPARTICLE IN THE PRESENCE OF A STATIONARY ZERO-MEAN STOCHASTIC SOURCE - MODEL FOR THERMAL-EQUILIBRIUM IN THE PAULTRAP

An analytical approach is developed to consider confined motion of a c harged microparticle within the Paul trap (an electrodynamic levitator trap) in an atmosphere near the standard temperature and pressure. Th e suggested approach is based on a second-order linear stochastic diff erential equation which describes dampled microparticle motion subject ed to the combined periodic parametric and random external excitations . To solve this equation a new ansatz is developed. This ansatz is a g eneralization of the Bogoliubov-Krylov decomposition technique, which is usually used to reduce the order of a differential equation. The so lution is obtained in the long time imaging limit by applying the Bogo liubov general averaging principle. In spite of the second-order form of the initial stochastic differential equation, the microparticle mot ion can be understood as a one-dimensional Markov process. Comparison in the long time imaging limit of the calculated data obtained from th e analytically derived expression for the standard deviation of confin ed microparticle stochastic motion with the experimentally obtained da ta demonstrates asymptotic agreement for regions where the dimensionle ss parameter kappa is much less than 1 (kappa less-than-or-equal-to 0. 005). Simple extremum analysis of the expression obtained for the stan dard deviation reveals that for the particular case of a large drag pa rameter alpha (alpha much greater than 8 square-root 12) there is a mi nimum in the standard deviation which is only a dependent.