Underdamped stochastic harmonic oscillator driven by Lévy noise

Area of Science:

• * Statistical physics
• * Nonlinear dynamics
• * Stochastic processes

Background:

• * The equipartition theorem is a fundamental concept in classical statistical mechanics.
• * Stochastic oscillators driven by non-Gaussian noise, such as Lévy noises, exhibit complex behaviors.
• * Understanding energy distribution in such systems is crucial for characterizing their nonequilibrium dynamics.

Purpose of the Study:

• * To investigate the distribution of potential and kinetic energy in stationary states of a linearly damped stochastic oscillator driven by Lévy noises.
• * To analyze the long-time behavior of energy distributions and their relation to the equipartition theorem.
• * To determine the influence of damping and Lévy noise parameters on energy partition.

Main Methods:

• * Analytical investigation of the stationary states of a damped stochastic oscillator.
• * Derivation and analysis of the probability distributions for kinetic and potential energies.
• * Examination of power-law asymptotics and deviations from the equipartition theorem.
• * Study of the impact of the damping coefficient and Lévy noise parameters (stability index α).

Main Results:

• * In the long time limit, kinetic and potential energy distributions follow power-law asymptotics and violate the equipartition theorem.
• * The damping coefficient dictates energy partition; higher damping favors potential energy storage.
• * A stochastic analog of the equipartition theorem is proposed for vanishing damping, where energy distributions share similar widths.
• * The ratio of instantaneous kinetic to potential energy exhibits universal power-law asymptotics, independent of Lévy noise parameters.
• * Lévy-stable fluctuations with stability index α<2 indicate a strongly nonequilibrium character.

Conclusions:

• * The damped stochastic oscillator driven by Lévy noises operates in a strongly nonequilibrium state.
• * Energy partition is governed by the interplay between damping and the nature of the driving noise.
• * The findings challenge classical equipartition theorem predictions in systems with Lévy-stable fluctuations.