  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.
