Nonergodic Brownian oscillator: low-frequency response

Abstract

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 ω 0 and the oscillator natural frequency exceeds a critical value ω 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 Ω lower than ω 0. The results complement those obtained earlier for the high-frequency response at Ω ⩾ ω 0 and require a different mathematical approach. The signature property of the high-frequency response is resonance when the external force frequency Ω coincides with the frequency of the localized mode ω ∗ . In the low-frequency domain Ω < ω 0 the condition of resonance Ω = ω ∗ cannot be met (since ω ∗ > ω 0 ). Yet, in the limits ω → ω c and Ω → ω 0 − , the oscillator shows a peculiar quasi-resonance response with an amplitude increasing with time sublinearly.