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arxiv: 2605.01723 · v1 · submitted 2026-05-03 · 🪐 quant-ph

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

Aref Pahlevani , Morteza Rafiee , Mehdi Ansari-rad This is my paper

Pith reviewed 2026-05-10 15:22 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Mpemba effectparametric drivingcoupled oscillatorscolored noiseanomalous relaxationcovariance matrixstability boundaryLorentzian noise
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The pith

Parametric driving near the stability boundary shortens the Mpemba crossing time in coupled oscillators, with colored noise expanding the region where the effect occurs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the Mpemba effect in two linearly coupled harmonic oscillators, one parametrically driven and connected to a thermal bath. It demonstrates that stronger parametric driving toward the instability limit causes hotter initial states to reach equilibrium faster than colder ones, measured by reduced crossing times. Colored noise, especially Lorentzian noise applied to both oscillators, further shortens these times and widens the range of drive and coupling strengths where the anomalous relaxation appears. The slow eigenvalue structure of the drift matrix sets the main behavior, while noise spectrum adjustments act mainly as a quantitative boost. This setup offers a controllable platform for studying how external driving influences relaxation pathways in noisy linear systems.

Core claim

In a pair of linearly coupled harmonic oscillators with one parametrically driven and coupled to an independent thermal bath, the Mpemba effect manifests as faster relaxation from higher initial temperatures. The covariance-matrix formalism yields the dynamics under white noise and Lorentzian colored noise in single- and dual-channel embeddings. Relaxation is tracked via Frobenius distance to steady state and projection onto the slowest mode. The crossing time decreases systematically as the parametric drive approaches the stability boundary, and dual-oscillator Lorentzian noise produces a stronger reduction while enlarging the parameter region of occurrence. The slow-mode structure of the 4

What carries the argument

Covariance-matrix evolution of the linear drift matrix under parametric modulation, with relaxation quantified by Frobenius norm distance and slowest-eigenmode projection.

If this is right

  • Increasing parametric drive strength toward the stability limit systematically reduces the Mpemba crossing time.
  • Dual-channel Lorentzian colored noise shortens the crossing time more than single-channel or white noise.
  • Colored noise enlarges the region in the drive-coupling plane where the Mpemba effect is observed.
  • The slow-mode structure of the drift matrix remains the dominant mechanism, with noise color providing secondary quantitative changes.
  • Both Frobenius and slowest-mode measures yield consistent evidence of the anomalous relaxation under the studied conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar parametric control might accelerate relaxation in other driven linear systems such as optomechanical or electrical resonators.
  • If the linear regime persists near the boundary, it suggests engineering faster equilibration in quantum open systems by approaching instability thresholds.
  • Tuning the noise spectrum could serve as an additional handle for enhancing anomalous effects in thermal relaxation problems.
  • The results motivate checking whether the same drive-induced speedup appears in nonlinear oscillator arrays or under quantum noise.

Load-bearing premise

The linear covariance description and chosen distance measures continue to capture the crossing accurately even as fluctuations grow large near the parametric stability boundary.

What would settle it

An experiment that tunes the parametric drive strength upward while tracking the relaxation crossing time and finds that the time stops decreasing or begins to increase before the stability boundary would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.01723 by Aref Pahlevani, Mehdi Ansari-rad, Morteza Rafiee.

Figure 1
Figure 1. Figure 1: FIG. 1. (color online) Time evolution of the Frobenius distance [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (color online) Projection amplitudes onto the slowest relax [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (color online) Time evolution of the Frobenius distance [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (color online) Mpemba strength, defined as [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (color online) Crossing time [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the Mpemba effect in two linearly coupled harmonic oscillators, one parametrically driven and coupled to independent thermal baths. Using the covariance-matrix formalism, it derives the relaxation dynamics for both Gaussian white noise and Lorentzian colored noise (single- and dual-channel embeddings). Relaxation is quantified via the Frobenius distance to steady state and projection onto the slowest mode of the drift matrix. The central results are that parametric driving near the stability boundary systematically reduces the Mpemba crossing time, that colored noise (especially dual-oscillator Lorentzian) further shortens crossing times and enlarges the parameter region exhibiting the effect, and that the slow-mode structure remains the dominant mechanism while noise color provides a secondary, quantitative correction.

Significance. Within the linear model the covariance equations are closed and exact, so the reported trends with drive strength and noise spectrum are mathematically reliable and constitute a clean demonstration that parametric driving can be used to tune anomalous relaxation. The comparison of noise embeddings and the identification of the slow-mode dominance are useful for designing experiments in driven mechanical or optomechanical systems. The work would be strengthened by explicit verification that the linear predictions remain qualitatively intact once weak nonlinearities are restored near the stability boundary.

major comments (2)
  1. The central claim that Mpemba crossing time decreases as the parametric drive approaches the stability boundary is load-bearing for the abstract and results. Within the linear time-periodic drift plus additive Gaussian noise the covariance ODE is exact, yet the steady-state covariance diverges as the largest Lyapunov exponent approaches zero. No section demonstrates that the reported Frobenius or slowest-mode trends survive the addition of even weak quartic restoring forces or amplitude-dependent damping, which become essential precisely in the regime where the headline effect is strongest.
  2. Abstract: the manuscript states that covariance-matrix equations are derived for white and colored noise, yet supplies neither the explicit matrix ODEs, their derivation steps, nor any error estimates or direct numerical checks of the crossing-time trends against stochastic simulations. This omission prevents independent verification of the quantitative statements about dual-channel Lorentzian noise producing stronger reduction than single-channel noise.
minor comments (2)
  1. The definitions of the two distance measures (Frobenius norm and slowest-mode projection) should be written explicitly, including the precise form of the projection operator onto the slowest eigenvector of the drift matrix.
  2. Figure captions and axis labels should state the precise values of the coupling strength, drive amplitude, and noise correlation time used in each panel so that the reported enlargement of the Mpemba region can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the linear-model results, and constructive suggestions. We address each major comment point by point below and indicate the revisions we will implement.

read point-by-point responses

  1. Referee: The central claim that Mpemba crossing time decreases as the parametric drive approaches the stability boundary is load-bearing for the abstract and results. Within the linear time-periodic drift plus additive Gaussian noise the covariance ODE is exact, yet the steady-state covariance diverges as the largest Lyapunov exponent approaches zero. No section demonstrates that the reported Frobenius or slowest-mode trends survive the addition of even weak quartic restoring forces or amplitude-dependent damping, which become essential precisely in the regime where the headline effect is strongest.

    Authors: We agree that the linear model becomes an approximation near the stability boundary, where the steady-state covariance diverges and nonlinear terms (quartic restoring forces or amplitude-dependent damping) would eventually dominate. Our manuscript focuses on the exactly solvable linear regime, in which the covariance equations are closed and the reported trends with drive strength are mathematically rigorous. We will add a new paragraph in the Discussion section that explicitly states this limitation, notes that the Mpemba trends hold exactly within the linear model, and suggests that weak-nonlinearity effects could be examined in future work via perturbation theory or direct stochastic simulations. We do not claim the trends survive arbitrary nonlinearities; the linear results remain useful for understanding the underlying slow-mode mechanism and for experimental regimes where nonlinearities remain perturbative. revision: partial

  2. Referee: Abstract: the manuscript states that covariance-matrix equations are derived for white and colored noise, yet supplies neither the explicit matrix ODEs, their derivation steps, nor any error estimates or direct numerical checks of the crossing-time trends against stochastic simulations. This omission prevents independent verification of the quantitative statements about dual-channel Lorentzian noise producing stronger reduction than single-channel noise.

    Authors: We apologize for the omission of the explicit equations. In the revised manuscript we will insert the full covariance-matrix ODEs for the white-noise case and for both single-channel and dual-channel Lorentzian embeddings, together with a concise derivation from the underlying stochastic differential equations. These will appear in the main text or a dedicated appendix. We will also add direct numerical comparisons between the covariance-matrix predictions and ensemble stochastic simulations for representative parameter sets, including error estimates on the crossing times and explicit confirmation that dual-channel Lorentzian noise yields a stronger reduction than the single-channel case. This will enable independent verification of the quantitative claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows standard linear stochastic dynamics

full rationale

The paper derives relaxation dynamics from the covariance-matrix ODE for a linear time-periodic drift matrix plus additive Gaussian noise (white or colored). This evolution equation is closed and exact by construction for the chosen model class, with all reported Mpemba crossing times obtained by direct integration or eigenmode analysis of that ODE. No parameter is fitted to data and then relabeled as a prediction, no ansatz is smuggled via self-citation, and the central trends (crossing time decreasing toward the stability boundary) are direct numerical consequences of the linear equations rather than tautological redefinitions. The approach is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard assumptions for linear stochastic differential equations and the operational definition of the Mpemba effect via crossing times; no free parameters or new entities are introduced in the abstract.

axioms (3)
  • domain assumption The system is a pair of linearly coupled harmonic oscillators with parametric driving on one and independent thermal baths.
    Stated as the physical setup under study.
  • domain assumption Relaxation dynamics are fully captured by the time evolution of the covariance matrix under white or Lorentzian colored noise.
    Invoked via the covariance-matrix formalism.
  • domain assumption The Mpemba effect is identified by a crossing in the relaxation curves measured by Frobenius distance or slowest-mode projection.
    Used to characterize anomalous relaxation.

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discussion (0)

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Reference graph

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