The Mpemba effect in the Descartes protocol: A time-delayed Newton's law of cooling approach
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arXiv
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2026
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| _version_ | 1866915923034636288 |
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| author | Santos, Andrés |
| author_facet | Santos, Andrés |
| contents | 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 $τ$, the waiting time $t_{\text{w}}$, and the normalized warm temperature $ω$, 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 $ω$ for given $τ$ and $t_{\text{w}}$. Within those bounds, the effect becomes maximal at a specific value $ω=\widetildeω(t_{\text{w}})$, and its magnitude is quantified by the extremal value of the temperature-difference function at this optimum. Accurate and compact approximations for both $\widetildeω(t_{\text{w}})$ and the maximal magnitude $\text{Mp}(t_{\text{w}})$ are derived, showing in particular that the absolute maximum at fixed $τ$ is reached for $t_{\text{w}}=τ$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03790 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Mpemba effect in the Descartes protocol: A time-delayed Newton's law of cooling approach Santos, Andrés Statistical Mechanics Materials Science Soft Condensed Matter Classical Physics 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 $τ$, the waiting time $t_{\text{w}}$, and the normalized warm temperature $ω$, 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 $ω$ for given $τ$ and $t_{\text{w}}$. Within those bounds, the effect becomes maximal at a specific value $ω=\widetildeω(t_{\text{w}})$, and its magnitude is quantified by the extremal value of the temperature-difference function at this optimum. Accurate and compact approximations for both $\widetildeω(t_{\text{w}})$ and the maximal magnitude $\text{Mp}(t_{\text{w}})$ are derived, showing in particular that the absolute maximum at fixed $τ$ is reached for $t_{\text{w}}=τ$. 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. |
| title | The Mpemba effect in the Descartes protocol: A time-delayed Newton's law of cooling approach |
| topic | Statistical Mechanics Materials Science Soft Condensed Matter Classical Physics |
| url | https://arxiv.org/abs/2602.03790 |