Mpemba Effect in Parametrically Driven Coupled Oscillators under White and Colored Noise

TL;DR

This paper investigates the Mpemba effect in coupled harmonic oscillators with parametric driving and noise, revealing how driving and colored noise influence anomalous relaxation times.

Contribution

It introduces a covariance-matrix formalism to analyze the Mpemba effect in driven oscillators under white and colored noise, highlighting control via parametric driving and noise characteristics.

Findings

Parametric driving reduces the Mpemba crossing time near stability boundary.

Colored Lorentzian noise enhances the Mpemba effect and expands its parameter region.

The slow-mode structure of the drift matrix is the main mechanism behind the effect.

Abstract

We study the Mpemba effect in a pair of linearly coupled harmonic oscillators, one of which is parametrically driven and coupled to an independent thermal bath. Using the covariance-matrix formalism, we derive the relaxation dynamics under both Gaussian white noise and Lorentzian colored noise, including single-channel and dual-channel noise embedding. We characterize relaxation through the Frobenius distance to the steady state and through the projection onto the slowest mode of the dynamical generator. Our results show that parametric driving provides the primary control knob for anomalous relaxation: as the drive approaches the stability boundary, the Mpemba crossing time decreases systematically. Colored noise further enhances the effect, with dual-oscillator Lorentzian noise producing a stronger reduction in the crossing time than single-oscillator noise and enlarging the parameter region where the Mpemba effect occurs. Nevertheless, the slow-mode structure of the drift matrix remains the dominant mechanism, while the influence of colored noise is secondary and mainly quantitative. We show that the Mpemba crossing time decreases as the system approaches the parametric stability boundary and that Lorentzian colored noise enlarges the region in the parametric and coupling strength plane where the Mpemba effect is observed.