Pareto fronts and trade-off relations from exact multi-objective optimization of thermal machines

Abstract

Thermal machines are physical systems that, when fueled by input energy, perform output tasks such as heat pumping or the production of work. Their performance is characterized with several, often competing quantities, such as power, efficiency, energy waste, and resilience to environmental noise. Multi-objective optimization provides a key tool to investigate the characterization of the best thermal machines operating in the irreversible linear-response regime. Here, we derive exact analytical parameterizations for the optimal (Pareto) fronts associated with any given choice of relative weights assigned to their mean extracted power $P$, efficiency $\eta$, entropy production $\Sigma$ and the amplitude of power fluctuations $\sigma^2_P$. The geometry of the front of endoreversible machines is universal: two-, three-, and four-objective trade-offs follow analytical formulae that do not depend on the value of any physical parameter of the machine. We show that such universal thermodynamic Pareto fronts also set quantitative fundamental limits for the performance of non-endoreversible machines. Furthermore, we demonstrate that our results apply to existing experimental data from different physical systems also beyond the linear regime, ranging from atomic to macroscopic scales, including single-atom engines, colloidal systems, macroscopic engines and power plants.