Optimal Control of a Mesoscopic Information Engine

Abstract

We analytically solve the finite-time control problem of driving an overdamped particle via an optical trap under costly measurement. By formulating this mesoscopic information engine within the Partially Observable Markov Decision Process (POMDP) framework, we demonstrate that the underlying Linear-Quadratic-Gaussian (LQG) dynamics decouple the optimal measurement and driving protocols. We derive the optimal feedback control law for the trap placement, which recovers the discontinuous Schmiedl-Seifert driving protocol in the open-loop limit and extends it to any measurement scheduling. For a costly, binary (on/off) sensor, we evaluate the optimal measurement protocol and derive physical bounds on the maximum gain that can be extracted from thermal fluctuations. We prove the emergence of deadline-induced blindness, a phenomenon where all measurements cease as the deadline approaches regardless of their cost. Taking the infinite-horizon limit, we find the exact periodic measurement schedules for the steady state as a function of the measurement cost $C$ and derive the macroscopic velocity envelopes beyond which viscous drag forces the engine into a net-dissipative regime. Finally, we generalize the results to a variable-precision sensor.

Figures (4)

• Figure 1: Optimal finite-time control dynamics of the information engine with binary measurements. (Top) Simulated spatial trajectory illustrating the true particle position ($x$, gray), the inferred belief mean ($\mu$, blue dashed) with $1\sigma$ uncertainty bounds (blue shaded), and the optimal deterministic trap placement ($\lambda$, red). Black markers denote discrete measurement events. (Bottom) Corresponding temporal evolution of the belief variance ($\Sigma$, purple). Thermal diffusion drives the uncertainty upward until it intersects the dynamically computed optimal threshold ($\Sigma_{th}(n)$, orange dashed), triggering a measurement that collapses the variance. As the deadline approaches ($t \to t_f$), the required threshold for profitable measurement rigorously diverges, demonstrating the onset of finite-time deadline blindness, here shown in grey when $\Sigma_{th}(n) > \Sigma_\kappa$. Simulation parameters: $k_B T=1$, $\kappa=1$, $\gamma=1$, $\Delta t=0.01$, $t_f=20$, $C=0.6 \, C_{max}$.
• Figure 2: Thermodynamic phase space of the information engine with binary measurements at steady-state. Heatmaps displaying the measurement frequency (top) and the thermodynamic viability ratio ($\mathcal{P}_{fluct}/\mathcal{P}_{drag}$) (bottom) as a function of macroscopic driving velocity $v$ and measurement cost $C$. The velocity envelope $C_{env}(v)$ (dashed line) separates the active, profitable engine regime from the net-dissipative regime. This operational space is bounded by two fundamental physical limits: the steady-state starvation threshold ($C = k_B T / 2$) at $v=0$, and the speed limit ($v = v_{max}$) at $C=0$, beyond which macroscopic viscous drag overwhelms the maximum extractable microscopic fluctuation power. Simulation parameters: $k_B T=1$, $\kappa=1$, $\gamma=1$.
• Figure 3: Optimal finite-time control dynamics of the information engine with variable-precision measurements. (Top) Simulated spatial trajectory illustrating the true particle position ($x$, gray) and the optimal deterministic trap placement ($\lambda$, red). The agent's belief mean is color-coded by the instantaneous Kalman gain $L_k$, showing the active measurement intensity. The shaded blue region denotes the $1\sigma$ posterior uncertainty. (Bottom) Corresponding temporal evolution of the belief variance ($\Sigma$, purple). Throughout the trajectory, the demon continuously measures with just enough precision to keep the variance locked to the dynamic target $\Sigma_{opt}^+$ (dashed orange). As the deadline approaches, the target diverges triggering the terminal "deadline blindness" phase (gray shading): the sensor is completely deactivated ($L_t \to 0$, visible as the dark shift in the belief mean colormap). Afterwards the variance converges to the thermal equilibrium variance $\Sigma_\kappa$ (dotted line). Simulation parameters: $k_B T=1$, $\kappa=1$, $\gamma=1$, $\Delta t=0.01$, $t_f=20$, $c=0.6 \, c_{max}$.
• Figure 4: Thermodynamic phase space of the information engine with variable-precision measurements at steady-state.. Heatmaps displaying the continuous precision rate ($\dot{\Sigma}_{meas}/\Sigma_\infty^{*2}$) (top) and the thermodynamic viability ratio ($\mathcal{P}_{fluct}/\mathcal{P}_{drag}$) (bottom) as a function of macroscopic driving velocity $v$ and the variable precision cost scalar $c$. The analytical envelope $c_{env}(v)$ (dashed line) separates the active, profitable engine regime from the net-dissipative regime. This operational space is bounded by two fundamental physical limits: the steady-state starvation threshold ($c = (k_B T)^2 / 2\kappa$) at $v=0$, and the speed limit ($v = v_{max}$) at $c=0$, beyond which macroscopic viscous drag overwhelms the maximum extractable continuous fluctuation power. Simulation parameters: $k_B T=1$, $\kappa=1$, $\gamma=1$.