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

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 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 w . Within those bounds, the effect becomes maximal at a specific value ω = ω ~ ( t w ) , and its magnitude is quantified by the extremal value of the temperature-difference function at this optimum. Accurate and compact approximations for both ω ~ ( t w ) and the maximal magnitude Mp ( t w ) are derived, showing in particular that the absolute maximum at fixed τ is reached for t 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.