- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Control protocols for harmonically confined run-and-tumble particles

Marco Baldovin* and Alessandro Manacorda

  • *Contact author: marco.baldovin@cnr.it
PDF

Phys. Rev. E 114, 014112 – Published 9 July, 2026

DOI: https://doi.org/10.1103/rspp-2334

Abstract

Run-and-tumble particles constitute one of the simplest models of self-propelled active matter, and provide an ideal playground for the understanding of out-of-equilibrium systems. We consider an idealized setup where one such particle is subject to a harmonic confining potential, and an external agent can vary in time the tumbling rate and the strength of the trap. We search for time-dependent control protocols steering the system between assigned end states, in a prescribed time interval. To this aim, we propose a description of the dynamics, alternative to the usual ones, in the form of an infinite set of ordinary differential equations. Solutions based on a suitable closure of such hierarchy, which we expect to hold true in the limit of long protocol duration, are discussed and compared with numerical simulations. We also look for the protocol completing the task with the minimal work, on average: the problem can be tackled analytically, again in the regime of slow (but not quasistatic) transformations. The solution provides insightful intuition on the optimal strategies for the control of active matter systems.

View figure in article

Physics Subject Headings (PhySH)

Collections

This article appears in the following collection:

Controlling Stochastic Dynamics Across Scales

We present a Collection of papers on Controlling Stochastic Dynamics Across Scales. It seeks to highlight novel studies on controlling the dynamics of complex stochastic systems with a rich phenomenology. Guest editors of the Collection are Étienne Fodor of the University of Luxembourg and Todd Gingrich of Northwestern University.

Article Text

References (43)

  1. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  2. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016).
  3. R. D. Leonardo, L. Angelani, D. Dell'Arciprete, G. Ruocco, V. Iebba, S. Schippa, M. P. Conte, F. Mecarini, F. D. Angelis, and E. D. Fabrizio, Bacterial ratchet motors, Proc. Natl. Acad. Sci. USA 107, 9541 (2010).
  4. P. Pietzonka, E. Fodor, C. Lohrmann, M. E. Cates, and U. Seifert, Autonomous engines driven by active matter: Energetics and design principles, Phys. Rev. X 9, 041032 (2019).
  5. T. Ekeh, M. E. Cates, and E. Fodor, Thermodynamic cycles with active matter, Phys. Rev. E 102, 010101(R) (2020).
  6. G. Gronchi and A. Puglisi, Optimization of an active heat engine, Phys. Rev. E 103, 052134 (2021).
  7. D. Mandal, K. Klymko, and M. R. DeWeese, Entropy production and fluctuation theorems for active matter, Phys. Rev. Lett. 119, 258001 (2017).
  8. P. Pietzonka and U. Seifert, Universal trade-off between power, efficiency, and constancy in steady-state heat engines, Phys. Rev. Lett. 120, 190602 (2018).
  9. I. Buttinoni, G. Volpe, F. Kümmel, G. Volpe, and C. Bechinger, Active Brownian motion tunable by light, J. Phys.: Condens. Matter 24, 284129 (2012).
  10. C. Maggi, F. Saglimbeni, M. Dipalo, F. De Angelis, and R. Di Leonardo, Micromotors with asymmetric shape that efficiently convert light into work by thermocapillary effects, Nat. Commun. 6, 7855 (2015).
  11. G. Vizsnyiczai, G. Frangipane, C. Maggi, F. Saglimbeni, S. Bianchi, and R. Di Leonardo, Light controlled 3D micromotors powered by bacteria, Nat. Commun. 8, 15974 (2017).
  12. G. Frangipane, D. Dell'Arciprete, S. Petracchini, C. Maggi, F. Saglimbeni, S. Bianchi, G. Vizsnyiczai, M. L. Bernardini, and R. Di Leonardo, Dynamic density shaping of photokinetic E. coli, eLife 7, e36608 (2018).
  13. L. K. Davis, K. Proesmans, and E. Fodor, Active matter under control: Insights from response theory, Phys. Rev. X 14, 011012 (2024).
  14. D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys. 91, 045001 (2019).
  15. T. Schmiedl and U. Seifert, Optimal finite-time processes in stochastic thermodynamics, Phys. Rev. Lett. 98, 108301 (2007).
  16. E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Optimal protocols and optimal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011).
  17. A. Baldassarri, A. Puglisi, and L. Sesta, Engineered swift equilibration of a Brownian gyrator, Phys. Rev. E 102, 030105(R) (2020).
  18. D. Lucente, A. Manacorda, A. Plati, A. Sarracino, and M. Baldovin, Optimal control of an electromechanical energy harvester, Entropy 27, 268 (2025).
  19. M. Baldovin, D. Guéry-Odelin, and E. Trizac, Shortcuts to adiabaticity for Lévy processes in harmonic traps, Phys. Rev. E 106, 054122 (2022).
  20. S. A. Loos, S. Monter, F. Ginot, and C. Bechinger, Universal symmetry of optimal control at the microscale, Phys. Rev. X 14, 021032 (2024).
  21. B. De Bruyne and F. Mori, Resetting in stochastic optimal control, Phys. Rev. Res. 5, 013122 (2023).
  22. R. Goerlich, T. D. Keidar, and Y. Roichman, Resetting as a swift equilibration protocol in an anharmonic potential, Phys. Rev. Res. 6, 033162 (2024).
  23. R. Goerlich, K. S. Olsen, H. Löwen, and Y. Roichman, Time-energy trade-off in stochastic resetting using optimal control, Phys. Rev. E 113, 014103 (2026).
  24. A. Prados, Optimizing the relaxation route with optimal control, Phys. Rev. Res. 3, 023128 (2021).
  25. N. Ruiz-Pino and A. Prados, Optimal control of uniformly heated granular fluids in linear response, Entropy 24, 131 (2022).
  26. M. Baldovin, D. Guéry-Odelin, and E. Trizac, Control of active Brownian particles: An exact solution, Phys. Rev. Lett. 131, 118302 (2023).
  27. A. G. Frim and M. R. DeWeese, Shortcut engineering of active matter: Run-and-tumble particles, arXiv:2304.06023.
  28. J. Schüttler, R. Garcia-Millan, M. E. Cates, and S. A. M. Loos, Active particles in moving traps: Minimum work protocols and information efficiency of work extraction, Phys. Rev. E 112, 024119 (2025).
  29. R. Garcia-Millan, J. Schüttler, M. E. Cates, and S. A. M. Loos, Optimal closed-loop control of active particles and a minimal information engine, Phys. Rev. Lett. 135, 088301 (2025).
  30. K. S. Olsen, R. Goerlich, Y. Roichman, and H. Löwen, Harnessing non-equilibrium forces to optimize work extraction, Nat. Commun. 16, 11031 (2025).
  31. M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis, Phys. Rev. E 48, 2553 (1993).
  32. J. Tailleur and M. E. Cates, Statistical mechanics of interacting run-and-tumble bacteria, Phys. Rev. Lett. 100, 218103 (2008).
  33. A. P. Solon, Y. Fily, A. Baskaran, M. E. Cates, Y. Kafri, M. Kardar, and J. Tailleur, Pressure is not a state function for generic active fluids, Nat. Phys. 11, 673 (2015).
  34. K. Malakar, V. Jemseena, A. Kundu, K. V. Kumar, S. Sabhapandit, S. N. Majumdar, S. Redner, and A. Dhar, Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension, J. Stat. Mech. (2018) 043215.
  35. M. Paoluzzi, A. Puglisi, and L. Angelani, Entropy production of run-and-tumble particles, Entropy 26, 443 (2024).
  36. M. Paoluzzi, A. Puglisi, and L. Angelani, Local entropy production rate of run-and-tumble particles, Phys. Rev. E 112, 054109 (2025).
  37. J. Tailleur and M. E. Cates, Sedimentation, trapping, and rectification of dilute bacteria, Europhys. Lett. 86, 60002 (2009).
  38. D. Guéry-Odelin, C. Jarzynski, C. A. Plata, A. Prados, and E. Trizac, Driving rapidly while remaining in control: Classical shortcuts from Hamiltonian to stochastic dynamics, Rep. Prog. Phys. 86, 035902 (2023).
  39. W. Mathews, M. Esrick, Z. Teoh, and J. Freericks, A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions, Condens. Matter Phys. 25, 33203 (2022).
  40. Y. Zhang and E. Fodor, Pulsating active matter, Phys. Rev. Lett. 131, 238302 (2023).
  41. A. Manacorda and E. Fodor, Diffusive oscillators capture the pulsating states of deformable particles, Phys. Rev. E 111, L053401 (2025).
  42. J. Sanders, M. Baldovin, and P. Muratore-Ginanneschi, Minimal work protocols for inertial particles in nonharmonic traps, Phys. Rev. E 111, 034127 (2025).
  43. M. Baldovin, Balbaldus/RnT-control: v1.0.0 (v1.0.0), Zenodo (2026), https://doi.org/10.5281/zenodo.20638509.

Outline

Information

Phys. Rev. E 114, 014112– Published 9 July, 2026
  • Received 4 February 2026
  • Accepted 26 May 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation