Redefining the Mpemba Effect： A Generalized Theory of History-Dependent Thermal Relaxation

Abstract

This study redefines the Mpemba effect as a manifestation of history-dependent thermal relaxation, moving beyond the static comparison of initial temperatures. A generalized cooling law, dTdt=−ueffTH, is proposed,where the effective cooling efficiency (ueff) depends on both temperature (T) and thermal history (H). Reanalysis of controlled experiments and molecular simulations shows that H imprints a persistent molecular memory that dictates ueff. Introduction of a state-space vector S=TueffT reveals a mathematical isomorphism with overdamped harmonic motion. This framework fundamentally explains the Mpemba effect: divergent cooling trajectories arise when identical T coexists with distinct ueff due to different H. The law’s universality is validated by its application to forensic post-mortem interval (PMI) estimation, correcting errors induced by pre-death fever or hypothermia, establishing the effect as a predictable feature of history-dependent systems.

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