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arXiv:2604.15790

Holographic Stirling engines and the route to Carnot efficiency

Nikesh Lilani, Manus R. Visser

hep-thcond-mat.quant-gasgr-qcquant-phSubmitted Apr 17, 2026• Announced Apr 20, 2026
v1 · Apr 17, 2026v2 · Aug 30, 2026
arXivHTMLPDF

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-Nordström 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.

Comments
49 pages, 8 figures

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Abstract We study the thermodynamic performance of a finite-time non-regenerative quantum Stirling-like cycle used as a heat engine. We consider specifically the case in which the working substance (WS) is a two-level system (TLS). The Stirling cycle is made of two isochoric transformations separated by a compression and an expansion stroke during which the WS is in contact with a thermal reservoir. To describe these two strokes we derive a non-Markovian master equation which allows to study the real-time dynamics of a driven open quantum system with arbitrary fast driving. Following the real-time dynamics of the WS using this master equation, the endpoints of the isotherms can deviate from the equilibrium thermal states. The role of this deviation in the performance of the heat engine is addressed. We found that the finite-time dynamics and thermodynamics of the cycle depend non-trivially on the different time scales at play. In particular, driving the WS at a time scale comparable to the resonance time of the bath enhances the performance of the cycle and allows for an efficiency higher than the efficiency of the quasistatic cycle, but still below the Carnot bound. However, by adding thermalization of the WS with the baths at the end of compression/expansion processes one recovers the conventional scenario in which efficiency decreases by speeding up the processes. In addition, the performance of the cycle is dependent on the compression/expansion speeds asymmetrically, which suggests new freedom in optimizing quantum heat engines. The maximum output power and the maximum efficiency are obtained almost simultaneously when the real-time endpoints of the compression/expansion processes are considered instead of the equilibrium thermal endpoint states. However, the net extractable work always declines by speeding up the drive.
In a quantum Stirling heat engine, the heat exchanged with two thermal baths is partly utilized for performing work by redistributing the energy levels of the working substance. We analyze the thermodynamics of a quantum Stirling engine operating in finite time. We develop a model in which a time-dependent potential barrier changes the energy-level structure of the working substance. The process takes place under a constant interaction with the thermal bath. We further show that in the limit of slow operation of the cycle and low temperature, the efficiency of such an engine approaches Carnot efficiency. We also show that the maximum output power, for the strokes that affect the energy levels, is obtained at an intermediate operating speed, demonstrating the importance of a finite-time analysis.
We investigate a quasi-static quantum Stirling cycle employing a spin-chain-star working fluid, comprising a central ancilla coupled to multiple Heisenberg chains via anisotropic exchange. A unitary transformation reduces the many-body thermodynamics to exact diagonalisation within a Hilbert space of dimension two to the power of the chain number plus one. We evaluate purely transverse, isotropic in-plane, and fully anisotropic configurations. The cycle exhibits tunable heat engine, refrigerator, and quantum accelerator regimes governed by ancilla frequency and coupling strength disparities. The transverse configuration demonstrates superior engine performance approaching the Carnot limit, corroborated by a closed-form partition function that permits the engine-refrigerator boundary to be evaluated exactly at arbitrary chain number. Conversely, the fully anisotropic architecture undergoes a genuine many-body crossover from engine to refrigerator with escalating chain multiplicity, driven by collective coupling amplification. Regenerator functionality proves non-ideal, rendering the cycle irreversible. Perfect regeneration emerges only along a precise quantum resonance locus where isochoric heat exchanges balance. Deviating from this condition exacerbates thermal imbalance with increasing system size, constituting the primary irreversibility source. Finally, kinematic coherence maximizes during cold-bath compression and amplifies with anisotropy and chain number. Originating from Hamiltonian basis rotation rather than non-equilibrium dynamics, this coherence remains dissociated from thermodynamic efficacy.
Abstract This research investigates the endoreversible quantum Stirling engine using the Bose-Einstein Condensate (BEC) as a working medium trapped in a generic law potential. The engine operates endoreversibly in order to capture a realistic condition, where the temperature of the working medium depends on the heating and cooling stroke times. The Fourier conduction law is applied to govern the rate of heat transfer between the system and the thermal reservoir in finite time, whilst in isoenergetic strokes, an infinite time is required to accomplish the thermal equilibrium. The results show that the condensed phase of BEC has the ability to enhance the efficiency of the engine, especially at temperatures near absolute zero. In conclusion, adjusting the temperature and volume ratio can optimize performance to achieve higher efficiency and power output of the Stirling engine.