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

Abstract

The catalytic extension of a discrete two-stroke engine employs a cyclic auxiliary system - the catalyst - that remains decoupled from the baths and performs no work, yet enhances power and efficiency beyond the corresponding non-catalytic counterpart. Theoretical models of discrete engines are relatively easy to analyze but remain challenging for experimental implementation due to the required control over individual strokes. In contrast, externally driven engines that are simultaneously coupled to both heat baths - the so-called continuous engines - are more experimentally feasible. Here, we establish an equivalence between discrete and continuous machines, both with and without a catalyst, by mapping the discrete unitary processes and thermalization steps onto an interaction Hamiltonian and a Markovian model of dissipation. As a result, by replacing probability flows with probability currents, we construct an analogous continuous machine corresponding to previously demonstrated catalytic schemes that generalize Otto engines. We illustrate this mapping for the simplest catalytic extension of the Otto engine, demonstrating catalytic enhancement in the continuous regime.

Figures (2)

• Figure 1: The upper panel illustrates a discrete Otto-like heat engine with a qudit catalyst. The left subpanel (A1) shows the work stroke, where the engine extracts work, while the right subpanel (A2) depicts the heat stroke, during which the qubits rethermalize to their initial temperatures, completing the cycle. The catalyst is chosen so that its marginal state remains unchanged after the work stroke, and correlations with the system are erased during the heat stroke, ensuring the engine is re-initialized. The lower panel (B) shows a continuous Otto-like engine assisted by a qudit catalyst. Unlike the discrete engine, it operates without distinct strokes, remaining continuously coupled to the driving field for work extraction and to the heat baths for thermalization. Dissipative interactions drive the engine toward a stationary state, resulting steady heat currents $J_h$, $J_c$, and power $P$.
• Figure 2: The power--efficiency trade-off for the continuous Otto engine and its catalytic extension with a two-dimensional catalyst, showing the full range of catalytic enhancement (cf. Ref. BiswasPRL). The engines operate at frequencies $\omega_h$ and $\omega_c$ with inverse temperatures $\beta_h$ and $\beta_c$. Dissipation is characterized by the equilibration time $\tau_{\rm eq}$ (identical for both baths) and driving rate $g$. The characteristic times are $\tau_{\rm Otto} = \tau_{\rm eq}\left(1 + 1/(g\tau_{\rm eq})^2\right)$ and $\tau_{\rm Catalytic} = A\,\tau_{\rm eq}\left(1 + B/(g\tau_{\rm eq})^2\right)$, with $A,B \le 1$ (cf. Eq. \ref{['char_time_ineq']}). Parameters: $\beta_h\omega_h=0.1$, $\beta_c/\beta_h=10$, $g\tau_{\rm eq}=10$, and $\eta_c = 1 - \beta_h/\beta_c$.