  Newtonian, undamped motion in single-well potentials belong to a class of
well-studied conservative systems. Here, we investigate and compare long-time
properties of fully deterministic motions in single-well potentials with
analogous randomized systems. We consider a special type of energy-conserving
randomization process: the deterministic motion is interrupted by hard velocity
reversals $\vec{v}(t_i)\to-\vec{v}(t_i)$ at random time instants $t_i$. In the
1D case, for fixed initial conditions, the differences in probability
distributions disappear in the long-time limit making asymptotic densities
insensitive to the selection of random time instants when velocity is reversed.
Substantially different probability distributions can be obtained, for
instance, through the additional randomization of initial conditions.
Analogously, in 2D setups, the probability distributions asymptotically are
insensitive to velocity reversals.
