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.