# Stationary engines in and beyond the linear response regime at the Carnot efficiency

## TL;DR

This paper investigates the conditions under which stationary engines can reach the Carnot efficiency, revealing different mechanisms for finite-size and macroscopic engines, and classifying these mechanisms using the tight-coupling window concept.

## Contribution

It identifies distinct conditions for Carnot efficiency attainment in finite-size versus macroscopic engines and introduces the tight-coupling window classification for these mechanisms.

## Key findings

- Finite-size engines' efficiency conditions differ from macroscopic ones.
- Tight-coupling condition implies Carnot efficiency beyond linear response for finite engines.
- Three mechanisms classified by tight-coupling window enable Carnot efficiency in different regimes.

## Abstract

The condition for stationary engines to attain the Carnot efficiency in and beyond the linear response regime is investigated. We find that this condition for finite-size engines is significantly different from that for macroscopic engines in the thermodynamic limit. For the case of finite-size engines, the tight-coupling condition in the linear response regime directly implies the attainability of the Carnot efficiency beyond the linear response regime. Contrary to this, for the case of macroscopic engines in the thermodynamic limit, there are three types of mechanisms to attain the Carnot efficiency. One mechanism allows engines to attain the Carnot efficiency only in the linear response limit, while other two mechanisms enable engines to attain the Carnot efficiency beyond the linear response regime. These three mechanisms are classified by introducing tight-coupling window.