A geometric bound on the efficiency of irreversible thermodynamic cycles*

Differential geometry provides powerful tools for understanding finite-time thermodynamics with notable successful applications in finding optimal protocols. Here, we use this approach to study closed thermodynamic cycles. We first focus on the Brownian Carnot engine and derive optimal geometric protocols for its finite-time operation. We find improvements over the current benchmarks in dissipated work, power, and efficiency over a wide range of experimentally accessible protocols durations. We also derive the optimal engine connecting the corners (by following the geodesics of the relevant inverse diffusion tensor) of the Carnot cycle, finding further improvements on dissipated work but at the cost of reduced power and efficiency. Building on these results, we next explore the full, unconstrained space of nonequilibrium thermodynamic cycles. We derive a novel geometric bound on the efficiency of any slowly-driven irreversible thermodynamic cycle and explicitly construct efficient heat engines operating in finite time that saturate this bound. Given the bound, these optimal cycles perform more efficiently than all other thermodynamic cycles operating as heat engines in finite time, including notable cycles such as those of Carnot, Stirling, and Otto. All of our predictions can be tested using existing experimental setups.