Energetics of the single-well undamped stochastic oscillators

TL;DR

This paper analyzes the energy behavior of non-harmonic, undamped stochastic oscillators driven by various noises, revealing how average energies depend on potential steepness and noise type, especially in long-time limits.

Contribution

It provides new analytical and numerical insights into how different noise types affect energy distributions in non-harmonic stochastic oscillators, extending understanding beyond Gaussian white noise.

Findings

Average total energy is insensitive to potential type under Gaussian white noise.

Long-time average energies grow as power-law with potential steepness.

Infinite potential well limit shows universal power-law energy growth.

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, $V(x) \propto x^{2n}$, under Gaussian white noise does not extend to other noise types. Nevertheless, in the long-time limit ($t \to \infty$), the average energies grow as power-law with exponents dependent on the steepness of the potential $n$. Another special limit corresponds to $n\to\infty$, 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.