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arxiv: 2604.15790 · v1 · submitted 2026-04-17 · ✦ hep-th · cond-mat.quant-gas· gr-qc· quant-ph

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Holographic Stirling engines and the route to Carnot efficiency

Nikesh Lilani , Manus R. Visser
Authors on Pith no claims yet

Pith reviewed 2026-05-10 08:06 UTC · model grok-4.3

classification ✦ hep-th cond-mat.quant-gasgr-qcquant-ph
keywords Stirling engineCarnot efficiencyregenerationholographic CFTAdS black holesheat capacitythermodynamic cycleconformal field theory
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The pith

Stirling engines reach Carnot efficiency when the working substance has volume-independent heat capacity at fixed volume.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that regenerative Stirling engines fall short of Carnot efficiency due to a net heat mismatch along the two constant-volume legs of the cycle. This mismatch disappears exactly when the heat capacity at fixed volume does not depend on volume, allowing perfect internal heat recycling and Carnot performance. The condition is satisfied by classical ideal gases and Van der Waals fluids but violated by quantum gases and thermal conformal field theories. For holographic CFTs dual to AdS-Schwarzschild and AdS-Reissner-Nordström black holes, exact efficiency formulas are derived, and efficiency approaches the Carnot value in the large-potential limit, with regeneration accelerating the approach.

Core claim

A sufficient condition for attaining Carnot efficiency in a reversible Stirling cycle, with or without regeneration, is that the fixed-volume heat capacity is independent of volume, which forces the isochoric heat mismatch to vanish. For thermal CFT states dual to AdS black holes, exact Stirling efficiencies are obtained, and in the fixed-potential ensemble these efficiencies asymptote to the Carnot value as the potential becomes large, with the approach becoming faster when regeneration is included.

What carries the argument

The isochoric heat mismatch, which measures the net heat that must be exchanged with external reservoirs after internal recycling and vanishes precisely when heat capacity at fixed volume is independent of volume.

If this is right

  • Classical ideal gases and Van der Waals fluids achieve Carnot efficiency in both regenerative and non-regenerative Stirling cycles.
  • Quantum ideal gases and CFT working substances have Stirling efficiencies strictly below Carnot because their heat capacities depend on volume.
  • Exact Stirling efficiency expressions exist for holographic CFTs dual to AdS-Schwarzschild and AdS-Reissner-Nordström geometries.
  • In the fixed-potential ensemble, regeneration reduces the gap to Carnot efficiency more rapidly than the non-regenerative case as potential grows.
  • The large-potential limit provides a concrete regime in which holographic engines can be made arbitrarily close to the Carnot bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The volume-independence condition may serve as a diagnostic for whether other thermodynamic cycles can be engineered to reach their theoretical efficiency bounds.
  • In analog gravity or condensed-matter systems that mimic holographic CFTs, one could test whether the approach to Carnot efficiency follows the same large-potential scaling.
  • The result suggests that regeneration is especially valuable for working substances whose heat capacities vary with volume, offering a practical route to improve real engines.

Load-bearing premise

The thermodynamic description of the cycle, including heat capacities and work terms obtained from the AdS/CFT dictionary, remains valid for the chosen equilibrium states throughout the engine cycle.

What would settle it

A direct calculation for a thermal CFT dual to an AdS black hole showing that Stirling efficiency does not approach the Carnot value in the large-potential limit would falsify the asymptotic claim.

Figures

Figures reproduced from arXiv: 2604.15790 by Manus R. Visser, Nikesh Lilani.

Figure 1
Figure 1. Figure 1: Comparison of Stirling efficiencies in the presence and absence of regeneration. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Operation of an alpha-type Stirling engine with a regenerator. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stirling cycle for a classical ideal gas. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: P V -diagrams for a Stirling cycle with a Van der Waals fluid working substance. This fluid has three qualitatively distinct isotherms in a P V -diagram: above, below and at the critical temperature Tcrit. Hence, there are 5 different ways in which one can operate the Stirling engine: (a) Th > Tcrit, Tc = Tcrit (b) Th = Tcrit, Tc < Tcrit (c) Th > Tc > Tcrit (d) Tc < Th < Tcrit (e) Th > Tcrit > Tc. We have … view at source ↗
Figure 5
Figure 5. Figure 5: Stirling cycle for a Bose-Einstein condensate. [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: P V -diagrams for holographic Stirling engines. D = 4 and k = 1 for these plots. potential Φ: ˜ S = 4πCxD−1 , V = Ωk,D−1R D−1 , (7.13) E = (D − 1)CxD−2 R  k + x 2 + α 2Φ˜ 2R 2  , (7.14) T = D − 2 4πRx  k + D D − 2 x 2 − α 2Φ˜ 2R 2  , (7.15) P = CxD−2 Ωk,D−1RD  k + x 2 + α 2Φ˜ 2R 2  , (7.16) Q˜ = 4(D − 2)CxD−2Φ˜R , (7.17) where α 2 = 2(D − 2) D − 1 , (7.18) C is the central charge in (7.2) and k is th… view at source ↗
Figure 7
Figure 7. Figure 7: T S-diagram of a Stirling cycle for a charged holographic CFT in the fixed Φ˜ ensemble. The corresponding diagram for an uncharged holographic CFT is qualitatively identical. D = 4 and k = 1 for this plot. As expected for a CFT, this depends only on the scale-invariant combinations T R and Φ˜R. Substituting into the expressions above and using R = R(V ) yields all thermody￾namic quantities as functions of … view at source ↗
Figure 8
Figure 8. Figure 8: Stirling efficiency as a function of the electric potential for charged holographic [PITH_FULL_IMAGE:figures/full_fig_p040_8.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript computes the efficiency of reversible Stirling engines (with and without regeneration) for classical fluids, quantum gases, Bose-Einstein condensates, and thermal CFTs dual to AdS-Schwarzschild and AdS-Reissner-Nordström black holes. It identifies a general sufficient condition for Carnot efficiency—volume-independent fixed-volume heat capacity that makes the isochoric heat mismatch vanish—and derives exact efficiency expressions for the holographic cases. In the fixed-potential ensemble these efficiencies are shown to approach the Carnot value in the large-potential limit, with regeneration producing a faster approach.

Significance. If the derivations hold, the work supplies a concrete holographic realization of a classical heat-engine cycle together with an explicit, falsifiable criterion (vanishing isochoric mismatch) for reaching the Carnot bound. The exact expressions for AdS black-hole duals and the controlled large-potential asymptotics constitute reproducible, parameter-free results that can be checked against standard black-hole thermodynamics; the regeneration mechanism is cleanly isolated as an internal heat-recycling effect.

minor comments (3)
  1. The abstract states that exact efficiency expressions are obtained for AdS-Schwarzschild and AdS-RN duals, yet the main text should include an explicit equation or subsection reference (e.g., §4.2, Eq. (27)) where these expressions are derived from the first law and the CFT equation of state.
  2. The fixed-potential ensemble is used for the asymptotic analysis; a brief remark clarifying how the chemical potential (or charge) is held fixed while the volume changes during the isochoric legs would remove any ambiguity in the cycle definition.
  3. A short table or plot comparing the Stirling efficiency versus the Carnot value for the listed working substances (ideal gas, Van der Waals, CFT) would make the central claim about the heat-mismatch mechanism immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive review of our manuscript. We appreciate the recognition of the significance of our results, including the general sufficient condition for Carnot efficiency via volume-independent fixed-volume heat capacity and the exact expressions for holographic CFTs. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from standard thermodynamics and AdS/CFT dictionary

full rationale

The paper applies the standard definitions of heat and work for a reversible Stirling cycle (with and without regeneration) to the thermodynamic potentials of various working substances. The sufficient condition for Carnot efficiency—that C_V independent of V implies vanishing isochoric heat mismatch—follows directly by integrating the heat capacity definitions over the isochoric legs; it is a mathematical identity, not a fitted or renamed result. Exact efficiency expressions for AdS-Schwarzschild and AdS-RN dual CFTs are obtained by substituting the known black-hole scaling relations (S ~ T^{d-1} or similar) into those cycle formulas. These scalings are prior, externally established results from black-hole thermodynamics and the AdS/CFT dictionary, not derived or fitted within the paper. No load-bearing step reduces a claimed prediction to a self-citation, an ansatz smuggled via citation, or a parameter fit renamed as output. The analysis therefore remains independent of the present work's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper rests on standard thermodynamic cycle analysis and the AdS/CFT correspondence without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption Thermodynamic relations for heat, work, and efficiency hold for the listed working substances in reversible cycles.
    Invoked throughout the efficiency calculations for all substances.
  • domain assumption AdS/CFT duality maps the thermodynamics of the boundary CFT to the bulk black-hole geometry for the engine states.
    Required for the holographic CFT results and exact expressions.

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