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

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.