Thermodynamic Cost of Regeneration in a Quantum Stirling Cycle

TL;DR

This work addresses whether regenerative quantum Stirling cycles can exceed the Carnot limit when regeneration costs are properly included. It develops a reduced open-system framework with a regeneration cost $W_{ ext{cost}}$ set by the Carnot heat-pump limit and analyzes two spin-$ frac{1}{2}$ working media, comparing to a conventional Stirling cycle. The main finding is that super-Carnot efficiencies disappear once the regeneration cost is accounted for; the modified efficiency $\\eta = W/(Q_h + W_{ ext{cost}})$ remains below the Carnot bound $\\eta_C = 1 - T_c/T_h$ but still exceeds the non-regenerative Stirling performance. An analytical upper bound for the conventional Stirling is derived via quantum relative entropy, and a sufficient condition on $W_{ ext{cost}}$ is given to guarantee $\\eta \\le \\\eta_C$ for the regenerative cycle. The authors also propose three quantum-regenerator models (non-Markovian reservoir, auxiliary quantum system, and collision-model regenerator) for future exploration, highlighting the importance of cost accounting for thermodynamic consistency in quantum heat engines.

Abstract

We study the regenerative quantum Stirling heat engine cycle within the standard weak-coupling, Markovian open quantum system framework. We point out that the regeneration process is not thermodynamically free in a reduced open-system description, and we treat the required work input as an explicit regeneration cost by modifying the cycle efficiency accordingly. We consider two working substances--a single spin-$1/2$ and a pair of interacting spin-$1/2$ particles--and investigate the cycle performance by taking the regeneration cost at its minimum value set by the Carnot heat-pump limit. For comparison, we also analyze the conventional Stirling cycle without regeneration under the same conditions. The super-Carnot efficiencies reported under the cost-free regeneration assumption disappear once the regeneration cost is included: the modified efficiency stays below the Carnot bound, while still remaining higher than the efficiency of the conventional Stirling cycle. For the conventional Stirling cycle, we provide a rigorous Carnot bound using quantum relative entropy, whereas for the regenerative cycle we derive a sufficient lower bound on the regeneration cost that guarantees thermodynamic consistency. Finally, we suggest three candidate quantum regenerator models for future work.

Figures (3)

• Figure 1: The schematic representation of the regenerative quantum Stirling cycle in the $U$ (internal energy) versus $\lambda$ (tuning parameter) diagram. The processes $A \rightarrow B$ and $C \rightarrow D$ are the quasistatic isothermal strokes performed at temperatures $T_h$ and $T_c$, respectively. The strokes $B \rightarrow C$ and $D \rightarrow A$ are two isochoric thermalization strokes performed at constant tuning parameters $\lambda_2$ and $\lambda_1$, respectively. Here, $Q_i$ ($i = 1, 2, 3, 4$) denotes the net heat exchange in each corresponding stroke of the cycle. The box under the cycle schematizes the regenerator. It stores the heat $|Q_2|$ released by the working substance during the $B \rightarrow C$ process and, during the $D \rightarrow A$ process, returns (all or part of) this stored heat to the system, thereby enabling heat regeneration and helping the cycle operate more efficiently. The arrow labeled $W_{\mathrm{cost}}$ indicates work input associated with this regeneration. Detailed explanations are given in the main text.
• Figure 2: Efficiency as a function of the relative magnetic field strength $\kappa$ ($\kappa=\lambda_1/\lambda_2$) for a single spin-$1/2$ working medium, with $\lambda_2=2.0$ and temperatures $T_h=3$ and $T_c=2$. The black solid curve shows the efficiency of the regenerative cycle without including any regeneration cost. The red dashed curve includes the regenerative cost $W_{\mathrm{cost}}$. The blue dot-dashed curve corresponds to the conventional Stirling cycle without a regeneration stage. The green dotted horizontal line marks the Carnot bound. We use dimensionless units where $\hbar=k_B=1$.
• Figure 3: Efficiency as a function of the interaction strength $J$ for a coupled-spins working medium, with temperatures $T_h=3$ and $T_c=2$, and tuning parameters $\lambda_1=2$ and $\lambda_2=1$. The line styles and color coding are the same as described in the caption of Fig. \ref{['fig2']}.