Energy-Efficient Control of Interacting Microscopic Systems: When Longer Paths Save Energy

Abstract

We experimentally and theoretically study the thermodynamically optimal control of interacting multiple-particle systems, focusing on collections of colloidal particles individually confined in optical traps. We investigate protocols that transport the system between prescribed trap configurations within a fixed time in the most energy efficient way. For Markovian systems with conservative pairwise interactions, we establish a general result in the low-noise limit: optimal particle trajectories are linear in space and time, corresponding to steady straight-line motion, irrespective of the specific interaction potential, even for nonlinear forces. Thus, conservative interactions do not modify the geometry of the optimal paths. This property breaks down in the presence of strong noise or nonconservative interactions. For the paradigmatic case of hydrodynamic coupling, we demonstrate experimentally that optimal control can involve curved trajectories that significantly reduce the energetic cost by exploiting collectively generated fluid flows. The emergence of curved paths as optimal solutions highlights a fundamental distinction between non-interacting and interacting systems and reveals a cooperative mechanism for energy-efficient control.

Figures (6)

• Figure 1: Sketch of the optimal control problem. Starting from an initial configuration with trap centers (blue $\times$) and particle positions (gray $\bigcirc$) at $\lambda_i^0$ and $r_i^0$, respectively, the control protocol $\lambda_i(t)$ (dashed black lines) translates the traps to their final positions $\lambda_i^\mathrm{f}$ (red $\times$) within a finite time $t_\mathrm{f}$. The particles position evolves according to the traps driving and thermal fluctuations (gray line); averaging over many realizations yields the mean particle trajectories (black lines). At $t=t_\mathrm{f}$, the particle positions generally do not coincide with $\lambda_i^f$ but subsequently relax to the new equilibrium positions without additional work.
• Figure 2: Example control problem with conservative particle interactions. (a) Sketch of the setup: two optical traps ($\{\bm \lambda_i\}$), each containing a particle ($\{\bm r_i\}$, circles), are driven anti-parallel from prescribed initial positions to final positions (crosses). The particles are coupled by a harmonic spring with rest length $l$ and stiffness $\Omega$. The gray circle of diameter $l$ marks the cross-over at which the spring force changes sign. (b-d) Optimal solutions for different rest lengths $l$. In all cases, the optimal protocols exhibit jumps (dashed lines) at the beginning ($\Delta_\mathrm{i}\bm \lambda_i$) and end ($\Delta_\mathrm{f}\bm \lambda_i$). The initial mean particle positions are obtained from the equilibrium distribution corresponding to the initial trap configuration. The black dot marks the symmetry center and center of mass of the system. As predicted by the general theoretical argument, the mean particle trajectories are straight lines in all cases, while the optimal protocols clearly deviate from straight-line motion for $l>0~\unit{\micro\meter}$. These curved protocols balance the nonlinear elastic coupling between the colloids. A finite noise strength (here zero) would not affect the solutions for $l=0~\unit{\micro\meter}$ (panel b), but for $l>0~\unit{\micro\meter}$ (panel c and d) it may induce nonlinear particle trajectories. Parameters: final time ${t_\mathrm{f}}=1~\unit{\second}$, trap coupling $\kappa=3~\unit{\micro\newton\per\meter}$, spring coupling $\Omega=2~\unit{\micro\newton\per\meter}$, vertical displacement of traps $H=4~\unit{\micro\meter}$, horizontal distance traps ${\Delta x}=3~\unit{\micro\meter}$, and friction constant $\gamma=1~\unit{\milli\pascal\second}$.
• Figure 3: (a,b) Mean force required to drag colloidal particles at constant velocity $v=3~\unit{\micro\meter\per\second}$ ($a=1.37~\unit{\micro\meter}$). The longitudinal force $F_\parallel$ is reduced when a second particle is dragged alongside at a separation of $8~\unit{\micro\meter}$, while the transverse force $F_\perp$ has zero mean. Vertical lines indicate theoretical predictions based on a fluid viscosity of $6.8~\unit{\milli\pascal\second}$cheng2008formulavolk2018density. Systematic deviations are attributed to particle-size variations and residual spatial inhomogeneities of the trap stiffness. (c,d) Color coded spatial dependence of the longitudinal and transverse components $\mu_{\parallel,\perp}$ of the simplified mobility tensor $\mathbf H$ [see Eq. \ref{['eq:Hsimple']}]. A force applied to the central particle (green arrow) induces a velocity of the second particle (black arrows). The reduction of $F_\parallel$ in (a) originates from longitudinal hydrodynamic coupling $\mu_\parallel$, whereas $\mu_\perp$ does not affect head-on motion, explaining the vanishing $F_\perp$. (e) Predicted optimal control protocols for two traps ($\kappa=3~\unit{\micro\newton\per\meter}$, $a=1.37~\unit{\micro\meter}$, $t_\mathrm{f}=5~\unit{\second}$). Blue: co-propagating configuration; red: counter-propagating configuration. Dashed lines show parabolic approximations parameterized by $\lambda_p$ [see Eq. \ref{['eq:lambda_parab']}]. (f) Measured average work $W$ as a function of $\lambda_p$ (open symbols), compared to simulations (solid lines). For a single particle, the straight path ($\lambda_p=0$) minimizes work. For co-propagating particles, the minimum shifts to $\lambda_p\simeq0.23\,a$ due to hydrodynamic force transfer, whereas for antiparallel driving $\lambda_p\simeq$$-0.44~a$ (the measured minima are marked as symbols at the bottom). Quantitative agreement is obtained by fitting the viscosity ($\eta_\rightrightarrows=6.0$, $\eta_{\leftrightarrows}=6.2$, $\eta_{\rightarrow}=6.9~ \unit{\milli\pascal\second}$), consistent with literature values. Residual discrepancies are attributed to particle polydispersity ($\sim4\%$), trap-stiffness variations ($\sim5\%$), and weak laser-induced heating.
• Figure 4: (a-d) Numerical prediction of optimal protocols for two, three, four and five traps that are shifted by 10 in $y$ direction and spacing of 6 on the $x$ axis. Viscosity and particle size are chosen as mentioned before in Fig. \ref{['fig:expres2d']}. The qualitative strategies of approaching traps remains true for all particle numbers. The average optimal work per particle $W$ normalized to that of an isolated particle $W^*_\mathrm{iso}$ decreases with particle number as shown in (e) for up to 10 particles. The amount of work individual traps contribute depending on the starting position $\lambda_{x}^0$ is shown in (f). Contributions from one group are connected with a spline as a guide to the eye. For any groups the traps in the center require the least amount of work while traps at the edge require the most. Interestingly, in larger groups also the amount of work required for the driving at the edge reduces with group size, which confirms the long range nature of hydrodynamic interactions.
• Figure 5: (a) Averaged particle trajectories in the $x$-dimension over time for $\lambda_p = 0.75~\unit{\micro\meter}$ for the co-propagating and $\lambda_p = -0.9~\unit{\micro\meter}$ for the counter-propagating configuration. Since protocol time is larger than the particle relaxation time in the trap, the particles on average closely follow the protocol shown as dashed black lines. (b) Difference between average particle position and trap center along the $x$-coordinate. Smooth and noisy lines correspond to simulations and averaged experimental data. The non-zero value for $t=0$ in the experimental data is due to remaining calibration errors and are close to the detection limit.