Holographic Stirling engines and the route to Carnot efficiency

A bstract 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, we identify a general sufficient condition for attaining Carnot efficiency: at fixed conserved charges, the fixed-volume heat capacity is independent of volume. This ensures that, at each temperature along the isochores, the heat released during cooling equals the heat absorbed during heating. An ideal regenerator can therefore store and return this heat reversibly, returning to its initial thermodynamic state with no net change in energy or entropy. 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-Nordström black holes, we obtain exact expressions for the Stirling efficiency. In the fixed-potential ensemble, both the regenerative and non-regenerative efficiencies approach the Carnot value in the large-potential limit. Regeneration reduces the leading deviation from the Carnot value, although both efficiencies have corrections of the same asymptotic order.