Nonergodic Brownian oscillator: Low-frequency response

TL;DR

This paper investigates the low-frequency response of a nonergodic Brownian oscillator, revealing a peculiar quasi-resonance behavior near critical spectral conditions, which complements earlier high-frequency response studies.

Contribution

It introduces a mathematical approach to analyze the low-frequency response of a nonergodic Brownian oscillator with a finite bath spectrum, highlighting quasi-resonance phenomena.

Findings

Low-frequency response exhibits quasi-resonance near critical spectral conditions.

Resonance occurs at high frequencies when external force matches localized mode frequency.

Amplitude increases sublinearly over time near critical points.

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 $\omega_*$. In the low-frequency domain $\Omega<\omega_0$ the condition of resonance $\Omega=\omega_*$ cannot be met (since $\omega_*>\omega_0$). 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.