  Trajectories of an overdamped particle in a highly unstable potential diverge
so rapidly, that the variance of position grows much faster than its mean.
Description of the dynamics by moments is therefore not informative. Instead,
we propose and analyze local directly measurable characteristics, which
overcome this limitation. We discuss the most probable particle position
(position of the maximum of the probability density) and the local uncertainty
in an unstable cubic potential, both in the transient regime and in the
long-time limit. The maximum shifts against the acting force as a function of
time and temperature. Simultaneously, the local uncertainty does not increase
faster than the observable shift. In the long-time limit, the probability
density naturally attains a quasi-stationary form. We explain this process as a
stabilization via the measurement-feedback mechanism, the Maxwell demon, which
works as an entropy pump. Rules for measurement and feedback naturally arise
from basic properties of the unstable dynamics. Observed thermally induced
effects are inherent in unstable systems. Their detailed understanding will
stimulate the development of stochastic engines and amplifiers and later, their
quantum counterparts.
