Brownian motion of a parametric oscillator driven by asymmetric square waves, classical fluctuation squeezing and unidirectional transport of matter in dense media

Brownian motion of a parametric oscillator with asymmetric square waves which take a spring constant during a given length of time and suddenly change to a different value with a different length of time-duration has been investigated analytically with a view to the Paul trap in contrast to usual symmetric fields. We assume that the square wave is periodically applied in time. The first and second moments for velocity and displacement have been analytically expressed in terms of simple matrices. The stable-unstable regions are shown explicitly. We consider how the unidirectional motion which is useful for the transport of particles may be recovered by the parametric oscillation. Moreover, it is shown that the fluctuation expressed by the mean square displacement becomes small in spite of increase of the amplitude of the square waves, which is called the classical fluctuation squeezing. This fluctuation squeezing is pronounced by the asymmetry of the waves and remarkably it takes double minima at certain regions. This model provides one of simple examples that randomness creates order.