Optimal transport of an active particle near a plane wall

Abstract

The control of active colloidal particles via optical traps is a cornerstone for research of matter at the micron and nanometer scale. A central challenge in this domain is the derivation of optimal transport protocols that minimize the mean work required to move a particle over a finite-time interval. Here we present a Ritz method in which open-loop protocols are constructed from a global basis of Chebyshev polynomials and optimised by a genetic algorithm. We apply the method to study optimal transport of an active particle, which is modelled as a force-dipole (or a stresslet) near a no-slip wall. The methodology is validated in the limits of zero activity and infinite wall separation, where it successfully recovers the known analytical protocols and the theoretical minimum work. Crucially, we demonstrate that the presence of the boundary breaks the time-reversal symmetry of the optimal protocol found in bulk solutions. This symmetry breaking is shown to be a complex function of the transport direction and the particle's intrinsic activity. Because the presented approach requires only the capability to simulate stochastic trajectories, it offers a robust, principled framework for optimizing transport protocols in complex fluid environments that remain inaccessible to exact analytical treatment.

Figures (9)

• Figure 1: Schematic of the system. An active colloidal particle (orange circle) is driven by a time-varying harmonic potential $V(\lambda(t))$. The particle exchanges energy with the trap, dissipates heat to the thermal bath, and self-generates drift through internal activity.
• Figure 2: Flowchart of the genetic algorithm. Each generation evaluates $\langle W \rangle$ via stochastic trajectory simulation, retains the top elites, and produces the next generation by tournament selection and annealed Gaussian mutation. The loop runs for $G=100$ generations.
• Figure 3: Optimal trap protocols for transport away from the wall, plotted as ${\Lambda}(t) - H_0$ versus normalised time $t/t_f$. (1) Columns: initial distance from wall $H_0$; rows: activity $\alpha$. (2) Solid orange: best GA protocol; shaded band: $\pm 1\sigma$ ensemble variance across 30 trials; dashed black: Seifert bulk prediction (Eq. \ref{['eq:seifert_protocol']} of appendix \ref{['app:seifert']}). (3) At $H_0 = 1000$ (rightmost column) the GA recovers the bulk solution for all $\alpha$. (4) Deviations from the bulk prediction grow as $H_0$ decreases and are modulated by activity.
• Figure 4: Heatmaps of GA optimisation results for away-from-wall transport. (a) Mean thermodynamic work $\langle W \rangle_{GA}$ as a function of $H_0$ and $\alpha$. (b) Percentage difference $\Delta W\% = 100\times(\langle W \rangle_{GA} - W_\text{Seifert})/W_\text{Seifert}$; negative values indicate that the GA protocol costs less work than the Seifert protocol under near-wall dynamics. The magnitude of $\Delta W\%$ is largest at small $H_0$ where wall effects are strongest.
• Figure 5: Symmetry breaking in the optimal protocol for a passive particle ($\alpha = 0$) at $H_0 = 2.0$. (a) Optimal away protocol. (b) Optimal towards protocol. Solid orange: best GA protocol; shaded band: $\pm 1\sigma$ ensemble variance across 30 trials; dashed black: Seifert bulk prediction. Fig. \ref{['fig:symmetry']}a exhibits a large initial jump and a plateau-then-rise shape; Fig. \ref{['fig:symmetry']}b tracks the bulk solution throughout most of the trajectory, deviating only near $t = t_f$.