Inverse engineering of cooling protocols: from normal behavior to Mpemba effects

TL;DR

This paper develops methods to engineer external cooling protocols that produce desired internal temperature evolutions, using models from phenomenological to microscopic, including effects like Mpemba phenomena.

Contribution

It introduces an inverse engineering approach to design external temperature protocols for targeted internal cooling trajectories, considering complex effects and model generalizations.

Findings

Analytical protocols for prescribed internal temperature evolution.

Backward-engineered protocols may not always exist or be unique.

Microscopic models confirm the feasibility of inverse cooling control.

Abstract

When a cup of hot coffee is suddenly put into a cold environment, it cools down as a function of time $t$ until the internal temperature $T_\text{int}$ of the coffee equals the external ambient temperature $T_\text{ext}$. This instantaneous shock-freezing corresponds to an imposed cooling protocol of the external temperature $T_\text{ext}(t)$, ideally described as a step-function in time, causing the time-dependent change of the internal temperature $T_\text{int}(t)$. While the effect of different given protocols $T_\text{ext}(t)$ on the resulting system cooling behaviour, embodied in $T_\text{int}(t)$, has been studied extensively, we consider here the inverse question: for a given system cooling $T_\text{int}(t)$ how can an appropriate protocol $T_\text{ext}(t)$ be engineered to produce the desired prescribed $T_\text{int}(t)$. We use both the phenomenological Newtonian equation for cooling and microscopic models, such as a discrete two-level system and a Brownian harmonic oscillator with time-dependent noise, to compute analytically the protocol $T_\text{ext}(t)$ needed to achieve a prescribed $T_\text{int}(t)$. We then discuss the same question for phenomenological generalizations of the Newtonian law which include anomalous Mpemba effects, overcooling, asymmetries in cooling and heating as well as delay phenomena. It is shown that backward-engineered protocols do not always exist and can be non-unique. The results are important for steering the cooling behavior by time-varying external heat sources in a systematic way.