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  • Open Access

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

Samuel Monter1, Lars T. Stutzer2, Sarah A. M. Loos2,3,*, and Clemens Bechinger1,4,†

  • *Contact author: sarah.loos@ds.mpg.de
  • †Contact author: clemens.bechinger@uni-konstanz.de
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Phys. Rev. X 16, 031079 – Published 30 September, 2026

DOI: https://doi.org/10.1103/cbbh-8r9c

Abstract

We experimentally and theoretically study the thermodynamically optimal control of interacting multiparticle 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 noninteracting and interacting systems and reveals a cooperative mechanism for energy-efficient control.

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Popular Summary

At microscopic scales, where thermal fluctuations and dissipation are unavoidable, finding control strategies that minimize energy consumption is a central challenge for the design and efficient operation of micromachines. Even at the single-particle level, with only one degree of freedom to control, optimal transport strategies can require complex control strategies. Here, we show that hydrodynamic interactions can be harnessed to control multiparticle systems more efficiently. Using optical tweezers to steer colloidal particles, and combining experiments with theory, we examine conditions in which particles are hydrodynamically coupled. The optimal trajectories then become curved and longer than the direct route between initial and final positions. Counterintuitively, these detours allow particles to exploit cooperative fluid motion generated by their neighbors, reducing viscous dissipation and lowering the required work by more than 30%.

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Phys. Rev. X 16, 031079– Published 30 September, 2026
  • Received 16 March 2026
  • Accepted 12 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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