Characterization and optimization of heat engines: Pareto-optimal fronts and universal features

TL;DR

This work develops a stochastic-thermodynamics framework for nanoscale heat engines described by discrete Markov dynamics in contact with two reservoirs, incorporating fluctuations and entropy production. It shows that the engine regime can be characterized by the simultaneous minima of the power-fluctuation $\mathrm{var}(\mathcal{P})$ and the entropy-production rate $\langle \dot{\sigma} \rangle$, and extends this view to a four-objective Pareto-front optimization balancing $\langle \mathcal{P} \rangle$, $\eta$, $\langle \dot{\sigma} \rangle$, and $\mathrm{var}(\mathcal{P})$ using the NSGA-II algorithm. The paper provides explicit results for two-state and three-state non-interacting systems and explores collective interacting systems, revealing how Pareto fronts transition from convex to locally concave as driving and parameter freedom increase, akin to protocol phase transitions. Collectively, the results connect thermodynamic uncertainty relations with multi-objective optimization, offering design principles for robustly optimized nanoscale heat engines and guiding future work on periodically driven and reservoir-asymmetric systems.

Abstract

Characterizing and optimizing nanoscopic heat engines require an appropriate understanding of the interplay between power, efficiency, entropy production and fluctuations. Despite significant recent advancements, including linear stochastic thermodynamics and thermodynamic uncertainty relations (TURs), a complete scenario remains elusive. In this work, we give a further step by showing that, under certain common and general conditions, the heat engine regime can be characterized by the minima of power fluctuations and entropy production, which together delimit its optimal performance, achieved when these conditions are fully satisfied. Conversely, when these conditions are not strictly met, the occurrence of the minimum still approximately describes the system, suggesting a broader range of applicability. Contrasting with most of studies in which the system optimization is carried out solely taking into account the power and efficiency, we introduce a multi-objective optimization framework based on Pareto fronts, also considering the role of fluctuation and dissipation. Our results reveal a general trend: while simultaneous optimization over a few parameters typically yields convex Pareto fronts, these fronts become concave as more parameters are varied freely and non-conservative driving becomes significant. Illustrating our findings, we consider simple two and three state systems as well as richer collective systems, exhibiting novel aspects of optimizations and protocol phase transitions.

Figures (6)

• Figure 1: Schematics of the driving operation for the three-state system in contact with the cold (left) and hot (right) thermal baths. Arrows denote the spin transitions $1\rightarrow 2\rightarrow 3\rightarrow 1$ favored in the clockwise (counterclockwise) by the non-conservative driving of strength $F$ provided through the coupling with the cold (hot) thermal baths. The protocol is similar for the collective system by replacing the sequence $(1\rightarrow 2\rightarrow3)$ to a local spin $s_i$ in which $(-1\rightarrow 0\rightarrow 1)$.
• Figure 2: Two-state thermodynamic observables and trade-offs. (a) Plot of the power, dissipation and power fluctuations as a function of efficiency $\eta$ for fixed $\beta_1 = 2$, $\beta_2 = 1$ and $\epsilon^{(2)} = -3.5$. Note that the heat engine regime, where $\langle\mathcal{P}\rangle <0$, is exactly bounded by both the minima of power fluctuations and of the dissipation; (b) pairwise power-efficiency Pareto front transitioning from the engine (black full line) to the heat pump (dashed line, red dots) regime at $\eta = \eta_C$ (red open triangle). Dots result from four-dimensional Pareto optimization. Inset: Pairwise optimization of $\langle \mathcal{P}\rangle$--$\eta/\eta_c$, only in the engine regime. (c)$\langle \mathcal{P}\rangle$--${\langle {\dot \sigma} \rangle}$ (red) and $\langle \mathcal{P}\rangle$-${\rm var}\mathcal{(P)}$ (blue) trade-offs. In panels (b, c), reservoir temperatures $\beta_1=2$, $\beta_2=1$ are fixed while $\epsilon^{(1)},\,\epsilon^{(2)} \in[-7,0]$) for the inset of panel (b), and $\beta_2 \in [0,\beta_1]$ with $\beta_1 = 2$. Open circles indicate the EMP in panel (b), and the DMP, FMP in panel (c).
• Figure 3: (a) Pairwise $\langle \mathcal{P}\rangle$--$\eta/\eta_c$ Pareto fronts for different non-conservative driving $F$'s (colored numbers). Optimizations have been carried out with respect to individual energies $\epsilon^{(\nu)}$'s and the temperatures $\beta_\nu$'s. Parameter ranges used in the optimization are $\epsilon^{(1)},\,\epsilon^{(2)} \in[-7,0]$, $\beta_1\in [0,2], \,\beta_2 \in [0,\beta_1]$. (b) Phase transitions in the order parameter $\phi=\langle {\cal P}\rangle/\langle {\cal P}\rangle_{\rm max}$ of the trade-offs in panel (a), for selected values of $F$.
• Figure 4: For the three-state system, the depiction of $\langle {\cal P}\rangle,\langle {\dot \sigma}\rangle$ and $\textrm{var}\,(\mathcal{P})$ for different set of individual energies. The energies were chosen as follows: $\Delta \epsilon_{ij}^{(\nu)}= (j-i)\,\epsilon^{(\nu)}$, where $\epsilon^{(2)} = -3.5$ (a); $\Delta \epsilon_{12}^{(2)} = \Delta \epsilon_{23}^{(2)} = \Delta \epsilon_{23}^{(1)} = -3.5$ (b); $\Delta \epsilon_{23}^{(1)} = (\beta_2/\beta_1) \Delta \epsilon_{23}^{(2)} = -3$ and $\Delta \epsilon_{12}^{(2)} = -3$ in (c) and $\Delta \epsilon_{12}^{(2)} = -3$, $\Delta \epsilon_{23}^{(1)} = -7$ and $\Delta \epsilon_{23}^{(2)} = -5$ (d). Parameters: $\beta_1 = 2$ and $\beta_2 = 1$.
• Figure 5: Depiction of $\langle {\cal P}\rangle,\langle {\dot \sigma}\rangle$ and $\textrm{var}\,(\mathcal{P})$ versus $\epsilon$ for different $\Delta$'s ($\Delta=10,5$ and $1$ from the left to right panels). The superscript 'eff' refers to the "effective" description, as discussed in Section \ref{['pfd']}. The horizontal black line at $\langle\mathcal{P}\rangle=0$ is shown for clarity, clearly showing the roots. Parameters: $F = 1.1$, $\beta_1 = 2$, and $\beta_2 = 1$.