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

TL;DR

The paper addresses how the Mpemba effect emerges in a time-delayed cooling setting by introducing the Descartes protocol, a three-reservoir scheme with single-step quenches. It develops an analytic solution to the time-delayed Newton's cooling equation, derives exact and approximate conditions for the direct and inverse Mpemba effects, and identifies an optimal waiting time $t_{\text{w}}^{\text{opt}}=\tau$ that maximizes the effect. The work provides compact, practical expressions for the optimal warm temperature $\tilde{\omega}(t_{\text{w}})$ and the maximal magnitude $\text{Mp}(t_{\text{w}})$, and compares the Descartes protocol to the two-reservoir setup, finding weaker maximal effects in the former. It also extends the analysis to finite-rate quenches, showing that strict Mpemba effects are prohibited by identical bath conditions but approximate effects persist for small bath-time scales $\sigma$. Overall, the framework offers a unified, analytically tractable approach to anomalous thermal relaxation in multi-step protocols and can be applied to other schemes such as the Pontus protocol.

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 $τ$, 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.

Figures (8)

• Figure 1: Schematic representation of three protocols applied to samples A and B. Panel (a) depicts the two-reservoir protocol of Refs. S24S25, where sample A undergoes a single-step quench at $t=-t_{\text{w}}$ and sample B a double-step quench at $t=-t_{\text{w}}$ and $t=0$. Panel (b) illustrates the Pontus protocol, in which three thermal reservoirs are used: sample B undergoes a single-step quench and sample A a double-step quench. Panel (c) shows the Descartes protocol, where three reservoirs are again used but both samples undergo single-step quenches at $t=-t_{\text{w}}$ (sample A) and $t=0$ (sample B).
• Figure 2: Normalized temperature $\theta(t)$ for different samples of type A and type B with a delay time $\tau=0.36$. In panel (a), the waiting time is $t_{\text{w}}=0.5$ for sample A, while the normalized warm temperature is $\omega=0.60$, $0.51$, $0.45$, and $0.30$ in samples B1, B2, B3, and B4, respectively. In panel (b), the normalized warm temperature is $\omega=0.60$ for sample B, while the waiting time is $t_{\text{w}}=0.50$, $0.40$, $0.35$, and $0.20$ in samples A1, A2, A3, and A4, respectively.
• Figure 3: Phase space for the direct Mpemba effect. In the top panels (a, b, c), the upper surface corresponds to the locus $\omega = \mathcal{E}(t_{\text{w}})$, while the lower surface represents $\omega = e^{-\kappa_0 t_{\text{w}}}$. The planes shown are (a) $\tau = 0.3$, (b) $t_{\text{w}} = 0.4$, and (c) $\omega = 0.4$. The region $\omega > \mathcal{E}(t_{\text{w}})$ is characterized by $\Delta(t \geq 0;t_{\text{w}}, \omega) < 0$, whereas $\Delta(t \geq 0;t_{\text{w}}, \omega) > 0$ holds in the region $\omega < e^{-\kappa_0 t_{\text{w}}}$. Consequently, a Mpemba effect occurs if and only if condition \ref{['Mp_cond']} is satisfied. Panels (d, e, f) show two-dimensional cross-sections of the three-dimensional phase space at $\tau = 0.3$, $t_{\text{w}} = 0.4$, and $\omega = 0.4$, respectively. Dashed lines indicate the loci where the Mpemba effect is maximal [see Sec. \ref{['sec2C']} and Eq. \ref{['MaximME']}].
• Figure 4: (a) Width $\delta\omega(t_{\text{w}})$ as a function of $t_{\text{w}}/\tau$ for $\tau=0.20$, $0.30$, and $0.36$. (b) Plot of $\delta\omega(t_{\text{w}}=\tau) = 1 - \tau - \kappa_0^{-1}$ and $\widetilde{\omega}(t_{\text{w}}=\tau)$ [see Sec. \ref{['sec2C']} and Eq. \ref{['owwApprox_tau']}] versus $\tau$.
• Figure 5: Plots of (a) the optimal normalized warm temperature $\widetilde{\omega}(t_{\text{w}})$, (b) the corresponding crossover time $\widetilde{t}_{\times}(t_{\text{w}})$, and (c) the magnitude of the Mpemba effect, $\text{Mp}(t_{\text{w}})$, all as functions of $t_{\text{w}}/\tau$ for $\tau = 0.20$, $0.30$, and $0.36$. In each panel, the dashed lines show the approximate results obtained from Eqs. \ref{['owwApprox']}, \ref{['otxA']}, and \ref{['MpApprox']}, respectively.