Optimal control strategy for collisional Brownian engines

TL;DR

The paper addresses finite-time optimization of a collisional Brownian engine driven by a time-periodic force within an underdamped Langevin framework. Using Euler–Lagrange variational calculus, it derives the exact optimal protocol, which comprises linear segments that maintain constant velocity interspersed with impulsive delta-like kicks that instantaneously reverse the velocity. This protocol yields closed-form expressions for mean input and output powers, efficiency, and entropy production, with the maximum-power condition and symmetry leading to a two-stroke cycle and near-unit efficiency at the power optimum, described by $\eta_{\rm opt} = 1 - \frac{8 \gamma}{4 \gamma + k \tau}$. Even when the delta impulses are smoothed to finite width, the engine retains high power, though efficiency and entropy production worsen, confirming the approach as a practical benchmark for finite-time optimization of Brownian engines.

Abstract

Collisional Brownian engines have recently gained attention as alternatives to conventional nanoscale engines. However, a comprehensive optimization of their performance, which could serve as a benchmark for future engine designs, is still lacking. In this work, we address this gap by deriving and analyzing the optimal driving protocol for a collisional Brownian engine. By maximizing the average output work, we show that the optimal protocol consists of linear force segments separated by impulsive delta-like kicks that instantaneously reverse the particle's velocity. This structure enforces constant velocity within each stroke, enabling fully analytical expressions for optimal output power, efficiency, and entropy production. We demonstrate that the optimal protocol significantly outperforms standard strategies (such as constant, linear, or periodic drivings) achieving higher performance while keeping entropy production under control. Remarkably, when evaluated using realistic experimental parameters, the efficiency approaches near-unity at the power optimum, with entropy production remaining well controlled, a striking feature of the optimal protocol. To analyze a more realistic scenario, we examine the impact of smoothing the delta-like forces by introducing a finite duration and find that, although this reduces efficiency and increases entropy production, the optimal protocol still delivers high power output in a robust manner. Altogether, our results provide a theoretical benchmark for finite-time thermodynamic optimization of Brownian engines under time-periodic drivings.

Figures (4)

• Figure 1: Schematic representation of the optimal Brownian engine (top) and the corresponding optimal protocol $F(t)$ (bottom). In the top panel, orange and black arrows indicate the direction of the velocity and the external driving force, respectively. The sharp transitions in steps II and IV correspond to delta-like kicks (amplified for better visualization) that instantaneously reverse the velocity, while steps I and III are the main operational stages. Explicitly, during step I, the protocol increases linearly, exactly compensating the natural acceleration to maintain a constant velocity; in step II, a sudden change reverses the velocity. In step III, the protocol decreases linearly to decelerate the particle while keeping its velocity constant; a final sudden change in step IV reverses the velocity again, preparing the system for the next cycle.
• Figure 2: Input work (red solid line), output work (black solid line), efficiency (blue dashed line), and entropy production (green dashed line) of the optimal strategy as functions of the driving period $\tau$. The parameters were set to $\gamma = 1$, $T = 1$, $v_1 = 1$ and $k = 10$ to enhance visual clarity.
• Figure 3: Panels (a), (b), and (c) show, respectively, the base-10 logarithm of the power, the efficiency, and the base-10 logarithm of the entropy production as functions of the driving period, for both the constant (red lines), optimal (black lines) and smoothed (green lines) protocols. The latter was computed using $d\tau = 0.01\,\tau$, and its results will be further discussed in Sec. \ref{['five']}. Resonance intervals are indicated by red dashed regions. All quantities were computed using the experimental parameters mentioned in the main text. Logarithmic scaling was used in panels (a) and (c) to enhance visualization and allow for a clearer comparison between the strategies.
• Figure 4: Performance of the engine as a function of the regularization width $d\tau$ of the delta contributions. (a) Output power $\overline{\mathcal{\dot{W}}}_{\rm{out}}$; (b) Log-10 of the preparation work $\overline{\mathcal{\dot{W}}}_{\rm{prep}}$; (c) efficiency $\eta$; (d) Log-10 of the entropy production $\overline{\sigma}$. The dashed regions in panels (b), (c), and (d) indicate the discontinuity in thermodynamic quantities occurring at $d\tau = 0$. We adopt the same parameters as in Fig. \ref{['compar']}, fixing the driving period at $\tau = 6\,\mu\text{s}$.