Energetics of single-well undamped stochastic oscillators

Abstract

The paper discusses analytical and numerical results for non-harmonic, undamped, single-well, stochastic oscillators driven by additive noises. It focuses on average kinetic, potential and total energies together with the corresponding distributions under random drivings, involving Gaussian white, Ornstein-Uhlenbeck and Markovian dichotomous noises. It demonstrates that insensitivity of the average total energy to the single-well potential type, V (x) ∝ x 2n , under Gaussian white noise does not extend to other noise types. Nevertheless, in the longtime limit (t → ∞), the average energies grow as power-law with exponents dependent on the steepness of the potential n. Another special limit corresponds to n → ∞, i.e. to the infinite rectangular potential well, when the average total energy grows as a power-law with the same exponent for all considered noise types.

Energy partition for anharmonic, undamped, classical oscillators

“…Analogous effect is observed for stochastic underdamped, undamped (frictionless), anharmonic oscillators [30] driven by the Gaussian white noise…”

“…De-pending on ν: potential (0 < ν < 2) or kinetic energy (ν > 2) energy can dominate, see Ref. [30]. Addition of linear damping to Eq.…”

Bounding energy growth in friction-less stochastic oscillators

“…Unbounded energy energy growth in the friction-less case For γ > 0 energy growth saturates at stationary value, which is proportional to the system temperature. In the friction-less case, i.e., for γ = 0 a different situation takes place and average energies grows in unbounded manner [26]. For n = 1 the system described by Eq.…”

“…Within current manuscript we study undamped (frictionless) motion in single-well potential of x 2n type under action of one or two Gaussian white noise sources. We continue research initiated in [25] and continued in [26] with the special attention to generalized equipartition relations [8] and mechanism of bounding energy growth based on stochastic resetting [13,21]. The model under study, basic theory and main results are presented in Sec.…”

Asymptotic Eigenvalues of Regular Sturm-Liouville Problems with Spectral Parameter-Dependent Boundary Conditions and Symmetric Single Well Potential

“…[5] and [8] have verified that the stationary states in symmetric single-well potentials can be characterized by more than two modal values. [27] discusses analytical and numerical results for nonharmonic, undamped, single-well, stochastic oscillators driven by additive noises. It focuses on average kinetic, potential, and total energies together with the corresponding distributions under random drivings, involving Gaussian white, Ornstein-Uhlenbeck, and Markovian dichotomous noises.…”