# Heating and cooling are fundamentally asymmetric

January 03, 2024

If you take a coin out of an ice bath, it warms up over time. Likewise, a hot coin that you have just taken out of a sauna cools down. The fact that systems, here the coin, thermally adapt to their environment is due to the heat flow that results from temperature differences. However, this adaptation process, called thermal relaxation, is much more complicated when large temperature changes bring a system far out of thermodynamic equilibrium.

As early as 2020, Aljaz Godec’s research group Mathematical bioPhysics at the Max Planck Institute (MPI) for Multidisciplinary Sciences in Göttingen predicted that small systems heat up faster than they cool down, provided that this happens under reasonable assumptions such as that the temperatures are chosen so that the cold and hot systems are equidistant from thermodynamic equilibrium (Phys. Rev. Lett. 125, 110602 (2020)). Accordingly, an ice-cold coin in the size of a few nano- or micrometers would heat up faster at room temperature than an equally small but hot coin would cool down. In collaboration with researchers led by Raul Rica from the Nanoparticles Trapping Laboratory at the University of Granada in Spain, the team now confirmed this prediction with experiments and by proving multiple of theorems. Thereby, they used an optically trapped colloidal particle. Remarkably, the physicists also show that the processes of heating and cooling are fundamentally different. This finding significantly changes the understanding of thermal relaxation.

According to the theory of “Linear Irreversible Thermodynamics”, developed by Lars Onsager (Nobel Prize 1968), there is a linear relationship between the flows and the corresponding thermodynamic driving forces for near-equilibrium systems at any point in time. Thus, systems relax quasi-statically in thermodynamic equilibrium by passing through a series of local equilibrium states – an assumption that is only justified a posteriori. The theory has strong implications, for example that (a) relaxation trajectories are uniquely specified by thermodynamic equilibria that interpolate between initial non-equilibrium and final equilibrium states, and (b) that a temperature at any point in time describes the state of the system (in terms of statistical mechanics). Furthermore, (c) a system started at a colder and a warmer temperature at which it is equidistant from thermodynamic equilibrium will reach equilibrium at the same rate and (d) the heating and cooling between two fixed temperatures will be symmetric.

The study by Godec and his team, now published in Nature Physics, shows that the statements (a)-(d) are false for systems far from equilibrium. To explain the observations and in particular the strong deviations from the prediction of linear irreversible thermodynamics, the researchers have developed a new theoretical framework called ‘Thermal Kinematics’. This framework introduces the concepts of a metric distance and a velocity in the abstract space of probability distributions. Among other things, the new theory reveals unforeseen discrepancies between the temperature difference, thermodynamic distance, and thermo-kinematic distance. If a system is prepared at two temperatures, one colder and one warmer than the environment, chosen so that the system is thermodynamically equidistant from equilibrium, the path taken by the system as it heats up is always longer than the path taken as it cools down. Nevertheless, because of the faster initial expansion phase, heating is faster than cooling – the micro-coin heats up faster than it cools down. The results are of great importance for applications around nanoscale energy conversion and thermal management of microscopic devices.

In addition to the systematic mathematical description and theoretical calculations that confirm the discovery, experiments in living cells also show that Godec and his team’s approach works. In the future, researchers could apply these new fundamentals in measurements with abstract dynamics in a high-dimensional potential and use them, for example, to study chemical reactions or protein or DNA folding.

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