Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when one of the heat baths it operates between is finite. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.
One immediate application of our theoretical results is to the study of irreversible processes in the nanoscale regime. In particular, we show how the amount of ordered energy needed to drive a small system out of equilibrium is larger than the amount one could obtain in a reverse process. This affects reversibility of thermodynamic cycles and, in turn, deteriorates performance of nanoengines. Despite these negative finite-size effects, we find that in specially engineered conditions nanoscale engines can still achieve the ultimate limit of efficiency.
Our results expand the realm of applicability of thermodynamics beyond the constraint of macroscopic systems, and thus provide new tools to study the universe at the smallest scale.
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