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Underdamped stochastic harmonic oscillator driven by Lévy noise.

Bartłomiej Dybiec1, Ewa Gudowska-Nowak1, Igor M Sokolov2

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This study explores energy distribution in damped stochastic oscillators driven by Lévy noises. Results show energy partition deviates from the equipartition theorem, with damping controlling the balance and indicating nonequilibrium behavior.

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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.