Nonergodic Brownian oscillator: Low-frequency response

Abstract

An undisturbed Brownian oscillator may not reach thermal equilibrium with the thermal bath due to the formation of a localized normal mode. The latter may emerge when the spectrum of the thermal bath has a finite upper bound ${\omega }_0$ and the oscillator natural frequency exceeds a critical value ${\omega }_c$, which depends on the specific form of the bath spectrum. We consider the response of the oscillator with and without a localized mode to the external periodic force with frequency $\Omega$ lower than ${\omega }_0$. The results complement those obtained earlier for the high-frequency response at $\Omega \ge {\omega }_0$ and require a different mathematical approach. The signature property of the high-frequency response is resonance when the external force frequency $\Omega$ coincides with the frequency of the localized mode ${\omega }_{\ast }$. In the low-frequency domain $\Omega <{\omega }_0$ the condition of resonance $\Omega ={\omega }_{\ast }$ cannot be met (since ${\omega }_{\ast }>{\omega }_0$). Yet, in the limits $\omega \to {\omega }_c$ and $\Omega \to {\omega }_0^{-}$, the oscillator shows a peculiar quasi-resonance response with an amplitude increasing with time sublinearly.

arXiv comment11 pages, 3 figures

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• Fractional oscillator
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• Nonergodic Brownian oscillator: High-frequency response