Memory-aware feedback enhances power in active information engines

TL;DR

The paper addresses whether information engines can reliably extract work in active baths that exhibit temporal memory. It introduces a memory-aware proportional feedback protocol in an overdamped harmonic trap and analyzes its performance through covariance-based relaxation dynamics, avoiding full memory erasure. The authors derive how work and power per cycle depend on the feedback gain, measurement noise, and bath activity, showing that intermediate gains can outperform full resetting and that memory can be exploited to boost performance in nonequilibrium environments. The findings provide guiding principles for designing high-performance information engines in active, memory-bearing surroundings and point to avenues for extending the framework to more general noise models and feedback strategies.

Abstract

We study an information engine operating in an active bath, where a Brownian particle confined in a harmonic trap undergoes feedback-driven displacement cycles. Unlike thermal environments, active baths exhibit temporally correlated fluctuations, introducing memory effects that challenge conventional feedback strategies. Extending the framework of stochastic thermodynamics to account for such memory, we analyze a feedback protocol that periodically shifts the potential minimum based on noisy measurements of the particle's position. We show that conventional feedback schemes, optimized for memoryless thermal baths, can degrade performance in active media due to the disruption of bath-particle memory by abrupt resetting. To overcome this degradation, we introduce a class of memory-preserving feedback protocols that partially retain the covariance between the particle's displacement and active noise, thereby exploiting the temporal persistence of active fluctuations. Through asymptotic analysis, we show how the feedback gain -- which quantifies the strength of positional shifts -- nontrivially shapes the engine's work and power profiles. In particular, we demonstrate that in active media, intermediate gains outperform full-shift resetting. Our results reveal the critical interplay between bath memory, measurement noise, and feedback gain, offering guiding principles for designing high-performance information engines in nonequilibrium environments.

Figures (5)

• Figure 1: (Color online) Operation of an information engine and memory effect in active baths. (a) Along the time line, measurement and feedback occur at $t = n\tau_-$ and $t = n\tau_+$, respectively. In the left panel (Measurement), the inverted triangle indicates the measured position $\hat{y}_{t_-}$ relative to the center of the potential. The red dashed line represents the particle density inferred from the measurement. In the right panel (Feedback), an instantaneous shift of the potential center changes the reference frame. (b) The left and right panels display contour plots of the joint probability distribution between the relative position $y$ and the active noise $\eta$ at $n \tau_-$ and $n \tau_+$, respectively. The distributions are evaluated in the periodic steady state. (c) The heatmap shows a joint probability distribution $p(y_t,\eta_t)$ obtained from numerical simulations using active Ornstein-Uhlenbeck (AOUP) process. The black dots and the connecting line represents the conditional average of the active noise given the position $y$.
• Figure 2: (Color online) Proportional adjustment as a protocol for memory preservation. (a) The inverted triangle indicates the measured position of the Brownian particle at time $t$, and the horizontal arrows denote the shift of the potential center, with magenta and blue vertical markers representing proportional adjustment and full shift, respectively. (b) Time evolution of the covariance $C_t$ between the relative position $y_t$ and the active noise $\eta_t$: blue for $g=1$ and magenta for $g=0.5$. The initial covariance is taken from the steady state of the cyclic operation, and increases as the proportionality $g$ deviates from 1. The black dashed line denotes the asymptotic covariance in the limit $\tau\to\infty$.
• Figure 3: (Color online) Profiles of average work extraction per cycle and its classification under varying control parameters. (a) Heatmap of the average work extracted per cycle as a function of the cycle period $\tau$ and feedback gain $g$. The black solid line in the right panel marks the zero-work boundary. The orange curve on each panel indicates the optimal feedback gain $g*$ for each value of $\tau$. As the cycle period shortens, $g*$ decreases monotonically below 1, deviating from the standard full-shift feedback (FSF). (b) Average extracted work per cycle as a function of $\tau$ for different combinations of feedback gain and measurement error: cyan, blue, olive-green, and magenta curves correspond to $(g,\sigma^2/V_\xi) = (0.5,~0.5)$, $(1.0,~0.5)$, $(1.0,~1.3)$, and $(1.0,~2.2)$, respectively, all in identical active baths. For the olive-green curve, the black vertical marker indicates the transition point where the average work changes sign, near $\tau \sim 0.66\tau_r$. Dashed lines indicate the asymptotic values as $\tau \to \infty$. (c) Classification of operational regimes in the $(g,\sigma^2)$ parameter space. The blue, magenta, and olive-green regions indicate parameter combinations where the average work is always positive, always negative, or changes sign as a function of $\tau$, respectively. The gray shaded region corresponds to settings where the engine either fails to satisfy the convergence condition of Eq. (\ref{['eq-CondSteady']}) or yields consistently negative work.
• Figure 4: (Color online) Profiles of power and classification under varying control parameters. (a) Power profiles are shown for different activity levels, defined by the dimensionless parameter $\alpha\equiv A^2/2\kappa k_{\text{B}}T$, represented by blue, orange, and green curves. Dashed lines indicate linear fits in the short-period limit ($\tau\ll\tau_r$), defined by the intercept $P^{(0)}$ and slope $P^{(1)}$. Each row shares a common active noise correlation time. The left column displays results for FSF, while the right column shows those for an attenuated feedback ($g=0.5$). Insets zoom into the short-period regime of the orange curves, and vertical black lines indicate the period $\tau$ at which the power is maximized. (b) All six panels include the same boundary curves, which demarcate regions where the short-period intercept $P^{(0)}$ is positive (lower left) or negative (upper right). Yellow curves indicate the activity-dependent boundary where the short-period slope $P^{(1)}$ changes sign, positive in the green/white regions and negative in the blue regions.
• Figure S1: (Color online) Local extrema of $\sigma_\alpha^{*2}(g)$. (a) Number of solutions $n(g_{\rm ext})$ to $f'_\alpha(g)=0$ as a function of $\alpha$ in the regime $g \in (0,2)$. Two separate transitions occur at $\alpha=4/9$ and $\alpha=1/2$. (b) Sign of $P^{(1)}$ at $\alpha=4/9$, analogous to Fig. \ref{['fig-maxpower_phase2']}(b). The yellow curve marks $P^{(1)}=0$, and the yellow cross ($\times$) indicates its slope vanishes. (c) Location of the extremum $g_{\rm ext}$ as a function of $\alpha$. At $\alpha = 4/9$, a single point bifurcates into a local minimum (blue) and a local maximum (red curve). (d) Sign of $P^{(1)}$ at $\alpha=0.49$. The yellow curve corresponds to $P^{(1)}=0$, showing both a local minimum and a local maximum. The blue (red) curve traces the trajectory of the minimum (maximum), starting from the black cross ($\times$) at $\alpha=4/9$. For $\alpha > 1/2$, the minimum disappears while the maximum becomes the global maximum, as illustrated in Fig. \ref{['fig-maxpower_phase2']}(b), particularly in the panels for $\alpha=1/2$, $0.52$, and $1$.