Anharmonic oscillator driven by additive Ornstein-Uhlenbeck noise

TL;DR

This paper analytically investigates a nonlinear oscillator driven by finite-correlation Ornstein-Uhlenbeck noise, revealing anomalous scaling, non-Boltzmann stationary distribution, and confirming results with simulations.

Contribution

It introduces a non perturbative analytical approach for small dissipation and finite correlation noise, extending understanding beyond previous large dissipation or white noise limits.

Findings

Physical observables grow algebraically with time.

Derived explicit stationary probability distribution function.

Results agree well with numerical simulations.

Abstract

We present an analytical study of a nonlinear oscillator subject to an additive Ornstein-Uhlenbeck noise. Known results are mainly perturbative and are restricted to the large dissipation limit (obtained by neglecting the inertial term) or to a quasi-white noise (i.e., a noise with vanishingly small correlation time). Here, in contrast, we study the small dissipation case (we retain the inertial term) and consider a noise with finite correlation time. Our analysis is non perturbative and based on a recursive adiabatic elimination scheme: a reduced effective Langevin dynamics for the slow action variable is obtained after averaging out the fast angular variable. In the conservative case, we show that the physical observables grow algebraically with time and calculate the associated anomalous scaling exponents and generalized diffusion constants. In the case of small dissipation, we derive an analytic expression of the stationary Probability Distribution Function (P.D.F.) which differs from the canonical Boltzmann-Gibbs distribution. Our results are in excellent agreement with numerical simulations.