Equivalence of Discrete and Continuous Otto-like Engines Assisted by Catalysts: Mapping Catalytic Advantages from the Discrete to the Continuous Framework

The catalytic extension of a discrete two-stroke engine employs a cyclic auxiliary system that remains thermally decoupled and performs no work, yet enhances power and efficiency beyond its noncatalytic counterpart. While discrete models are analytically tractable, they remain experimentally challenging due to the precise control required over individual strokes. Continuous engines, simultaneously coupled to heat baths and a driving field, offer a more experimentally feasible alternative. Here, we establish a unified framework connecting discrete and continuous machines-with and without a catalyst-by mapping unitary strokes and thermalization steps onto an interaction Hamiltonian and Markovian dissipation. Replacing probability flows with probability currents yields a continuous analog of catalytic Otto-like schemes. Focusing on the simplest catalytic extension, we demonstrate that catalytic enhancement persists in the continuous regime. Our results demonstrate how discrete models offer valuable insights for analyzing the performance of continuous thermal machines.