Steady-state heat engines driven by finite reservoirs

TL;DR

This work addresses the realism gap in stochastic heat engines by modeling finite-size reservoirs that are themselves coupled to infinite baths. Using a two-scenario framework—Scenario I with cyclic, finite-reservoir interactions and Scenario II with simultaneous coupling—the authors derive an effective inverse temperature (entropic temperature) that governs entropy production and show how finite heat capacity shifts reservoir temperatures and thus thermodynamic fluxes. They analyze a minimal two-state engine and a minimal three-state interacting engine, demonstrating that finite reservoirs can both enhance power (in some cases) and enlarge the heat-pump operating regime, while also revealing that the Curzon-Ahlborn efficiency bound can be surpassed under finite-reservoir conditions. The results provide quantitative tools and optimization insights for nanoscale heat engines operating under realistic thermal isolation and finite reservoir heat capacities, with implications for device design and energy conversion at the micro- and nano-scale.

Abstract

We provide a consistent thermodynamic analysis of stochastic thermal engines driven by finite-size reservoirs, which are in turn coupled to infinite-size reservoirs. We consider a cyclic operation mode, where the working medium couples sequentially to hot and cold reservoirs, and a continuous mode with both reservoirs coupled simultaneously. We derive an effective temperature for the finite-size reservoirs determining the entropy production for two-state engines in the sequential coupling scenario, and show that finite-size reservoirs can meaningfully affect the power when compared to infinite-size reservoirs in both sequential and simultaneous coupling scenarios. We also investigate a three-state engine comprising two interacting units and optimize its performance in the presence of a finite reservoir. Notably, we show that the efficiency at maximum power can exceed the Curzon-Ahlborn bound with finite reservoirs. Our work introduces tools to optimize the performance of nanoscale engines under realistic conditions of finite reservoir heat capacity and imperfect thermal isolation.

Figures (9)

• Figure 1: Average power $\langle\mathcal{P}\rangle$ (solid lines) and entropy production rate $\langle\dot{\sigma}\rangle$ (dashed lines) as a function of $\varepsilon_2 - \varepsilon_1$ in Scenario I, for a two-state engine placed in contact with a cold finite-size reservoir and a hot infinite-size reservoir. We consider that the finite-size reservoir is characterized by (a) a constant heat capacity and (b) an ensemble of two level systems. Insets: Power versus $\varepsilon_2-\varepsilon_1$ for different values of $\tau$. In (c), we plot $\langle\mathcal{P}\rangle$ (solid lines) and $\langle\dot{\sigma}\rangle$ (dashed lines) as a function of $\varepsilon_2 - \varepsilon_1$ in Scenario II.
• Figure 2: Efficiency heat maps for distinct values of $\kappa_1$ and $\kappa_2$ in Scenario II. Red circles mark the crossover from heat engine to heat pump regimes through the ideal efficiency, $\eta/\eta_c=1$. Parameters: $T_2=2T_1=1$ and $V_1=0.2$.
• Figure 3: Optimized power and efficiencies as a function of $\kappa_2$ for $\kappa_1\rightarrow \infty$ in Scenario II. Panels (a) and (b) depict the maximum efficiency $\eta_{mE}$ and the efficiency at maximum power $\eta_{mP}$ for $V_2/V_1=3$ and $V_2/V_1=2.5$, respectively. Insets: The depiction of $\eta^{\star}_{mP}$ and $\langle\mathcal{P}\rangle_{mP}$ (right) versus $\kappa_2$, respectively.
• Figure A1: Schematics of dynamics and finite thermal reservoirs for the two-state system in scenario I.
• Figure A2: Schematics of dynamics and finite thermal reservoirs for a generic setup in scenario II.