Mpemba meets Newton: Exploring the Mpemba and Kovacs effects in the time-delayed cooling law

Despite extensive research, the fundamental physical mechanisms underlying the Mpemba effect, a phenomenon where a substance cools faster after initially being heated, remain elusive. Although historically linked with water, the Mpemba effect manifests across diverse systems, sparking heightened interest in Mpemba-like phenomena. Concurrently, the Kovacs effect, a memory phenomenon observed in materials such as polymers, involves rapid quenching and subsequent temperature changes, resulting in nonmonotonic relaxation behavior. This paper probes the intricacies of the Mpemba and Kovacs effects within the framework of the time-delayed Newton's law of cooling, recognized as a simplistic yet effective phenomenological model accommodating memory phenomena. This law allows for a nuanced comprehension of temperature variations, introducing a delay time (τ) and incorporating specific protocols for the thermal bath temperature, contingent on a defined waiting time (t_{w}). Remarkably, the relevant parameter space is two-dimensional (τ and t_{w}), with bath temperatures exerting no influence on the presence or absence of the Mpemba effect or on the relative strength of the Kovacs effect. The findings enhance our understanding of these memory phenomena, providing valuable insights applicable to researchers across diverse fields, ranging from physics to materials science.