Nonequilibrium energetics of sensing and actuation by a smart active particle

Abstract

Smart active agents must allocate finite energetic resources across distinct functions, yet the underlying thermodynamic trade-offs remain poorly understood. Here, we introduce a minimal model of a self-steering particle with an internal polarity-cue sensor coupled to an external environmental field, decomposing its steady-state entropy production rate into locomotion, actuation, and sensing costs. This separation exposes an energetic bookkeeping structure underlying even the simplest form of embodied navigation. The emergence of Pareto fronts linking energetic expenditure to localisation precision and path-following performance shows that feedback-controlled active motion is constrained by quantitative thermodynamic bounds that persist across distinct task geometries.

Figures (5)

• Figure 1: (a) Schematic illustration of the model of a smart active Brownian particle with explicit sensing and actuation. The particle self-propels at constant speed along its instantaneous heading direction $\hat{\bm u}(t)$ (blue vector). An internal polarity sensor with readout $\hat{\bm u}_\varphi(t)$ (green vector) aligns stochastically with a position-dependent steering policy $\hat{\bm u}^*({\bm r})$ (red vector), encoding information about the external environment. The heading direction is in turn actively steered toward the sensor readout, closing a feedback loop between sensing and actuation. (b) Exemplary particle trajectory (black) in the point-target setup, where the steering policy $\hat{\bm u}^*(r)$ (red vector field in the background) points toward a fixed target position denoted by the yellow star. At selected points along the trajectory, the three unit vectors shown in panel (a) --- the heading direction $\hat{\bm u}(t)$, the sensor readout $\hat{\bm u}_\varphi(t)$, and the steering policy $\hat{\bm u}^*({\bm r})$ --- are displayed. The trajectory was obtained from numerical integration of the Langevin equations with parameters $\tilde{\mu}=50$, $\tilde{\kappa}=1$, $\sigma_\varphi^2=0.01$, and ${\rm Pe}=10$.
• Figure 2: Dissipation-accuracy tradeoff for an active Brownian motion with polarity sensing and actuation tasked with localisation at a point target. (a) Numerical results for the entropy production and variance of the radial displacement for different choices of $(\tilde{\kappa},\sigma_\varphi^2)$ (grey scatter) are shown alongside the numerical (orange) and analytical (black dashed) Pareto fronts. Here, we set ${\rm Pe}=10$. (b) Relative contribution of the partial entropy production terms identified in Sec. \ref{['sec:epr_compact']} to the total dissipation (excluding self-propulsion) along the Pareto front.
• Figure 3: Localisation within a target disc. (a) Thermodynamic costs of actuation, (green), and sensing (orange), as functions of the renormalised alignment strength $\tilde{\kappa}$. Dashed lines indicate the theoretical predictions [cf. (\ref{['eq:sigma_sens_reg']}-\ref{['eq:div_ustar_reg']})] while symbols show values obtained from numerical simulations of Eqs. \ref{['eq:governing_langevin']}. (b) same as (a) for the total entropy production (purple), and variance of the particle displacement from the target center (blue). (c) The stationary density $\rho(r)$ profile obtained from Eq. \ref{['eq:rho_part_reg']} (dashed lines) and Langevin dynamics simulations (solid lines). In all panels, $R=0.1$, $\tilde{\mu}=50$, $\sigma_\varphi^2=0.01$, and ${\rm Pe}=10$.
• Figure 4: Navigation along a target path. (a) Comparison between theory (dashed curves) and numerical simulations (symbols) for the total entropy production (purple) and of the variance in the transverse direction (blue) are shown as functions of $\tilde{\kappa}$ and for three distinct values of $\varepsilon$. (b) Probability density functions of the orthogonal displacement obtained from the numerical simulations (solid curves) compared to those predicted by Eq. \ref{['eq:rho_path']} (dashed curves) for different values of $\tilde{\kappa}$ (left) and $\varepsilon$ (right). The close match demonstrates the quantitative accuracy of the theoretical framework also in this setup.
• Figure 5: Pareto fronts for navigation along a target path obtained by substituting the density \ref{['eq:rho_path']} into the Eqs. (\ref{['eq:epr_act_pinned']}--\ref{['eq:epr_sens_fast']}) and \ref{['eq:mean_v']}. The resulting integral expressions are evaluated numerically. To reduce computational costs, we work in the weak steering regime $\tilde{\kappa} \ll 1$ by only retaining corrections of leading order in $\tilde{\kappa}_{\rm eff}$ in the relevant expressions. Comparing across different values of the confinement length scale $\varepsilon$, we observe that both the precision-dissipation (a) and speed-dissipation (b) tradeoffs are softened for increasing $\varepsilon$.