Beyond the thermodynamic limit: finite-size corrections to state interconversion rates

TL;DR

This paper develops a mathematical framework to analyze thermodynamic transformations of finite-size systems, bridging single-shot and asymptotic thermodynamics, and explores implications for heat engines and work extraction.

Contribution

It introduces second-order asymptotics for finite-size thermodynamics, providing precise relations between interconversion rates and infidelity, extending thermodynamic theory beyond the thermodynamic limit.

Findings

Finite-size effects cause irreversibility in thermodynamic cycles.

A precise expression for the gap between distillable work and work of formation.

Finite heat baths affect heat engine performance, with conditions where finite-size effects vanish.

Abstract

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.