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Anharmonic Oscillator Driven by Additive Ornstein–Uhlenbeck Noise

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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 (PDF) which differs from the canonical Boltzmann–Gibbs distribution. Our results are in excellent agreement with numerical simulations.

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Authors and Affiliations

  1. Service de Physique Théorique, Centre d’Études de Saclay, 91191, Gif-sur-Yvette, Cedex, France

    Kirone Mallick

  2. Institut de Recherche sur les Phénomènes Hors Équilibre, Université de Provence, 49 rue Joliot-Curie, BP 146, 3384, Marseille Cedex 13, France

    Philippe Marcq

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  1. Kirone Mallick
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  2. Philippe Marcq
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Correspondence to Philippe Marcq.

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Mallick, K., Marcq, P. Anharmonic Oscillator Driven by Additive Ornstein–Uhlenbeck Noise. J Stat Phys 119, 1–33 (2005). https://doi.org/10.1007/s10955-004-2135-5

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  • DOI: https://doi.org/10.1007/s10955-004-2135-5

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