  Dynamical stability is a prerequisite for control and functioning of desired
nano-machines. We utilize the Caldeira-Leggett master equation to investigate
dynamical stability of molecular cogwheels modeled as a rigid, propeller-shaped
planar rotator. In order to match certain expected realistic physical
situations, we consider a weakly nonharmonic external potential for the
rotator. Two methods for investigating stability are used. First, we employ a
quantum-mechanical counterpart of the so-called "First passage time" method.
Second, we investigate time dependence of the standard deviation of the rotator
for both the angle and angular momentum quantum observables. A
perturbation-like procedure is introduced and implemented in order to provide
the closed set of differential equations for the moments. Extensive analysis is
performed for different combinations of the values of system parameters. The
two methods are, in a sense, mutually complementary. Appropriate for the short
time behavior, the First passage time exhibits a numerically-relevant
dependence only on the damping factor as well as on the rotator size. On the
other hand, the standard deviations for both the angle and angular momentum
observables exhibit strong dependence on the parameter values for both short
and long time intervals. Contrary to our expectations, the time decrease of the
standard deviations is found for certain parameter regimes. In addition, for
certain parameter regimes nonmonotonic dependence on the rotator size is
observed for the standard deviations and for the damping of the oscillation
amplitude.
