Holographic Stirling engines and the route to Carnot efficiency

TL;DR

This paper analyzes the efficiency of reversible Stirling engines across various substances, identifying conditions for Carnot efficiency and providing exact results for holographic CFTs related to black hole thermodynamics.

Contribution

It introduces a comprehensive framework for evaluating Stirling engine efficiency with diverse quantum and classical substances, highlighting the role of heat capacity and regeneration.

Findings

Regeneration improves efficiency by internal heat recycling.

Carnot efficiency is achieved when heat capacity is volume-independent.

Exact efficiency expressions are derived for holographic CFTs dual to black holes.

Abstract

We compute the efficiency of the reversible Stirling engine, with and without regeneration, for a broad class of working substances including Van der Waals fluids, quantum ideal gases (Bose and Fermi), Bose-Einstein condensates, thermal conformal field theories (CFTs), and holographic CFTs. Regeneration acts as an internal heat recycling mechanism that enhances efficiency by reducing the net heat exchange with external reservoirs. For regenerative Stirling cycles, a central role is played by the intrinsic heat mismatch between the two isochoric branches, which controls the deviation of the efficiency from the Carnot bound and quantifies the extent to which internally exchanged heat can be perfectly recycled. We identify a general sufficient condition for attaining Carnot efficiency, namely that the fixed-volume heat capacity is independent of the volume, ensuring that the isochoric heat mismatch vanishes. While this condition is satisfied for classical ideal gases and Van der Waals fluids, it is violated for quantum ideal gases and CFT working substances. For thermal CFT states dual to AdS-Schwarzschild and AdS-Reissner-Nordstr\"{o}m black holes we obtain exact expressions for the Stirling efficiency. In the fixed-potential ensemble, we show that the Stirling efficiency asymptotes to the Carnot value in the large-potential limit, with a faster approach in the presence of regeneration.