Attainability of Carnot efficiency with autonomous engines.

The maximum efficiency of autonomous engines with a finite chemical potential difference is investigated. We show that, without a particular type of singularity, autonomous engines cannot attain the Carnot efficiency. This singularity is realized in two ways: single particle transports and the thermodynamic limit. We demonstrate that both of these ways actually lead to the Carnot efficiency in concrete setups. Our results clearly illustrate that the singularity plays a crucial role in the maximum efficiency of autonomous engines.