The Mpemba effect: hot water can freeze faster than cold water under certain conditions. The phenomenon has been observed experimentally but lacks a universal macroscopic theory. Newtonian cooling — where heat loss is proportional to the temperature difference from the environment — predicts the opposite: a hotter system always takes longer to equilibrate because it starts farther from the target.
Lin, Tu, and Ma propose a mechanism grounded in linear irreversible thermodynamics. Their generalized cooling law includes a memory term: the cumulative heat exchange Q(t), integrated over the system's entire cooling history, feeds back into the instantaneous cooling rate. A system that has shed more total heat cools faster not because of its current temperature but because of its thermal history. The hotter system has exchanged more heat by any given time, and this accumulated exchange accelerates its subsequent relaxation.
The model introduces two coefficients. M controls the memory enhancement: when positive, it amplifies the Mpemba effect; when negative, it can produce an inverse Mpemba effect where the cooler system equilibrates faster. I controls asymptotic behavior and can prevent complete thermalization in some regimes. The interplay between these terms generates a family of anomalous relaxation behaviors from a single framework.
The through-claim is about what the system knows. Newtonian cooling is memoryless — the rate depends only on the current state. The generalized law adds history. The system's past thermal trajectory modifies its present dynamics. A hotter system doesn't cool faster because it's currently hotter. It cools faster because it has been cooling from a higher starting point for its entire history, and that history has changed something — the structural evolution of the system, the heat flux pathways, the internal configuration — in a way that accelerates what comes next.