# Deterministic and randomized motions in single-well potentials

## TL;DR

This paper compares long-term behaviors of deterministic and randomized motions in single-well potentials, revealing that in the long run, probability distributions become insensitive to velocity reversals under certain conditions.

## Contribution

It introduces a specific energy-conserving randomization method involving velocity reversals and analyzes its effects on long-term probability distributions in 1D and 2D systems.

## Key findings

- Distributions become insensitive to velocity reversals in 1D long-term limit.
- Additional randomization of initial conditions affects distributions.
- In 2D, distributions are also insensitive to velocity reversals asymptotically.

## Abstract

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.