Deterministic and randomized motions in single-well potentials

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 v(t i ) → − 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.

Energy partition for anharmonic, undamped, classical oscillators

“…(2) and (3). Here, using the probabilistic approach [26], we calculate the energy partition for any V (x) ∝ |x| ν (ν > 0) potential.…”

“…Maxima of the p(v) density are located at points where the acceleration is minimal, i.e., the velocity is maximal [26]. Formulas for p(x) and p(v) densities, see Eqs.…”

“…Consequently, in order to calculate the energy partition, it is not possible to use the methodology applied in equation ( 2) and ( 3). Here, using the probabilistic approach [26], we calculate the energy partition for any…”

“…The probability density p(x) has maxima at points where the particle spends most time, see equation (7), i.e., at points where the velocity is minimal [26]. These points correspond to reversal points, i.e., to the points where the velocity smoothly changes its sign.…”

“…Maxima of the p(v) density are located at points where the acceleration is minimal, i.e., the velocity is maximal (ν > 1) or minimal (ν < 1) [26]. Formulas for p(x) and p(v) densities, see equations ( 8) and ( 12), can be used to derive main results of the current research, i.e., time averaged energies given by equations ( 13) and ( 14) along with appropriate variances, see equations ( 16) and ( 17), and formulas for moments of the arbitrary order which are given by formulas (18) and (19).…”