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arxiv: 2602.03790 · v2 · submitted 2026-02-03 · ❄️ cond-mat.stat-mech · cond-mat.mtrl-sci· cond-mat.soft· physics.class-ph

The Mpemba effect in the Descartes protocol: A time-delayed Newton's law of cooling approach

Andr\'es Santos This is my paper

Pith reviewed 2026-05-16 07:14 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.mtrl-scicond-mat.softphysics.class-ph
keywords Mpemba effectDescartes protocolNewton's law of coolingthermal relaxationtime delayquench protocolsanomalous cooling
0
0 comments X

The pith

The Descartes protocol yields exact bounds on warm temperature for the Mpemba effect under delayed Newton's cooling.

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

The paper introduces the Descartes protocol, a three-reservoir scheme in which samples experience single-step quenches at staggered times, to study the direct and inverse Mpemba effects with a time-delayed version of Newton's law of cooling. This setup cleanly separates the delay time τ, the waiting time t_w, and the normalized warm temperature ω. For instantaneous quenches the authors derive exact bounds on ω (given τ and t_w) inside which the Mpemba effect appears, locate the value ω̃(t_w) that maximizes its strength, and supply compact approximations showing that the largest effect at fixed τ occurs when t_w equals τ. They also compare the maximal magnitude to an earlier two-reservoir protocol and show that an approximate Mpemba effect survives for finite-rate quenches when the bath timescale is short enough.

Core claim

For instantaneous quenches, the Mpemba effect exists inside explicit bounds on the normalized warm temperature ω for any chosen delay τ and waiting time t_w; within those bounds the effect reaches its peak at a specific ω = ω̃(t_w) whose magnitude is given by the extremal value of the temperature-difference function, and the absolute maximum over all choices at fixed τ is attained when t_w = τ. Compact approximations for both ω̃(t_w) and the peak magnitude Mp(t_w) are obtained. The three-reservoir Descartes protocol produces a smaller maximal effect than a two-reservoir scheme despite the extra control parameter. For finite-rate quenches strict equality of the final bath conditions rules out

What carries the argument

The Descartes protocol: a three-reservoir thermal scheme in which each sample undergoes a single-step quench at different times, governed by the time-delayed Newton's law of cooling.

Load-bearing premise

The time-delayed Newton's law of cooling accurately captures the thermal relaxation dynamics in the three-reservoir Descartes protocol for both instantaneous and finite-rate quenches.

What would settle it

Record the temperature-difference function versus ω at fixed τ and t_w; the central claim is refuted if the function lacks a maximum at the predicted ω̃(t_w) or if the location of that maximum fails to satisfy the derived bounds.

Figures

Figures reproduced from arXiv: 2602.03790 by Andr\'es Santos.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of three protocols applie [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Normalized temperature [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase space for the direct Mpemba effect. In the top pa [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Width [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots of (a) the optimal normalized warm temperature [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Ratio between the magnitude of the Mpemba effect i [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Schematic representation of the Descartes heati [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
read the original abstract

We investigate the direct and inverse Mpemba effects within the framework of the time-delayed Newton's law of cooling by introducing and analyzing the Descartes protocol, a three-reservoir thermal scheme in which each sample undergoes a single-step quench at different times. This protocol enables a transparent separation of the roles of the delay time $\tau$, the waiting time $t_{\text{w}}$, and the normalized warm temperature $\omega$, thus providing a flexible setting to characterize anomalous thermal relaxation. For instantaneous quenches, exact conditions for the existence of the Mpemba effect are obtained as bounds on $\omega$ for given $\tau$ and $t_{\text{w}}$. Within those bounds, the effect becomes maximal at a specific value $\omega=\widetilde{\omega}(t_{\text{w}})$, and its magnitude is quantified by the extremal value of the temperature-difference function at this optimum. Accurate and compact approximations for both $\widetilde{\omega}(t_{\text{w}})$ and the maximal magnitude $\text{Mp}(t_{\text{w}})$ are derived, showing in particular that the absolute maximum at fixed $\tau$ is reached for $t_{\text{w}}=\tau$. A comparison with a previously studied two-reservoir protocol reveals that, despite its additional control parameter, the Descartes protocol yields a smaller maximal magnitude of the effect. The analysis is extended to finite-rate quenches, where strict equality of bath conditions prevents a genuine Mpemba effect, although an approximate one survives when the bath time scale is sufficiently short. The developed framework offers a unified and analytically tractable approach that can be readily applied to other multi-step thermal protocols.

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

0 major / 3 minor

Summary. The manuscript introduces the Descartes protocol, a three-reservoir thermal scheme analyzed via the time-delayed Newton's law of cooling, to study direct and inverse Mpemba effects. For instantaneous quenches it derives exact bounds on the normalized warm temperature ω (for fixed delay time τ and waiting time t_w) that permit the effect, identifies an optimal ω̃(t_w) that maximizes the magnitude, supplies compact approximations for both ω̃(t_w) and the maximal magnitude Mp(t_w), shows that the absolute maximum at fixed τ occurs at t_w = τ, compares the maximal magnitude unfavorably with a previously studied two-reservoir protocol, and extends the analysis to finite-rate quenches where a genuine Mpemba effect is precluded but an approximate version survives when the bath timescale is short.

Significance. If the derivations hold, the work supplies an analytically tractable, parameter-separated framework for the Mpemba effect that yields exact bounds, an explicit optimum, and a clear statement that the largest effect occurs at t_w = τ. The direct comparison with the two-reservoir case and the finite-rate caveat are useful for assessing protocol dependence. The approach is readily extensible to other multi-step protocols and therefore constitutes a modest but concrete advance in the modeling of anomalous thermal relaxation.

minor comments (3)
  1. [Approximations section] The abstract states that 'accurate and compact approximations' are derived for ω̃(t_w) and Mp(t_w); the manuscript should state the error bounds or the range of τ and t_w over which these approximations remain accurate (e.g., relative error < 1 %).
  2. [Comparison paragraph] In the comparison with the two-reservoir protocol, the precise parameter values (τ, t_w, ω) used for the magnitude comparison should be listed explicitly so that the claim of a 'smaller maximal magnitude' can be reproduced.
  3. [Finite-rate quenches] The finite-rate extension concludes that 'strict equality of bath conditions prevents a genuine Mpemba effect'; a short sentence clarifying what 'strict equality' means in terms of the bath temperature schedule would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful and accurate summary of our manuscript, their positive assessment of its significance, and the recommendation for minor revision. The referee correctly identifies the key results on exact bounds for the Mpemba effect, the optimal waiting time t_w = τ, the compact approximations, and the comparison to the two-reservoir protocol. We have prepared the revised version incorporating any minor clarifications.

Circularity Check

0 steps flagged

No significant circularity: derivations follow directly from model equations

full rationale

The paper's central results—exact bounds on ω for the Mpemba effect, the optimal value ω̃(t_w), and the statement that the absolute maximum occurs at t_w=τ—are obtained as direct analytical consequences of solving the time-delayed Newton's law differential equations for the three-reservoir Descartes protocol. No parameters are fitted to data and then relabeled as predictions, no ansatz is smuggled via self-citation, and no uniqueness theorem or self-referential definition is invoked to force the outcomes. The comparison to a prior two-reservoir protocol is a secondary observation rather than a load-bearing premise, and the finite-rate extension explicitly notes the absence of a genuine Mpemba effect. The derivation chain is therefore self-contained against the model's own equations without reduction to its inputs by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the time-delayed Newton's law governs the temperature evolution in the protocol. No new particles or forces are introduced. The parameters τ, t_w, and ω are control variables rather than fitted constants.

free parameters (3)
  • delay time τ
    Control parameter defining the time shift between quenches in the protocol.
  • waiting time t_w
    Control parameter for the interval after the first quench.
  • normalized warm temperature ω
    Control parameter whose bounds determine existence of the Mpemba effect.
axioms (1)
  • domain assumption Temperature evolution obeys the time-delayed Newton's law of cooling.
    The entire analysis is built on this differential-equation model.

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