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Optimal work control for an odd-diffusive tracer under active fluctuations

Koushik Goswami1,*, Hong-Yan Shih (施宏燕)1,2,†, Cheng-Hung Chang (張正宏)1,3, Hsuan-Yi Chen (陳宣毅)1,2,4, and Pik-Yin Lai (黎璧賢)1,4,‡

  • *Contact author: koushik.goswami@phys.ncts.ntu.edu.tw
  • †Contact author: hongyan@as.edu.tw
  • ‡Contact author: pylai@phy.ncu.edu.tw
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Phys. Rev. Research 7, 043309 – Published 17 December, 2025

DOI: https://doi.org/10.1103/qrvl-mng6

Abstract

We investigate optimal work protocols for a tracer particle with odd mobility, confined in a rotationally asymmetric parabolic potential, and driven by temporally correlated (active) noise originating from an active process. Two control strategies are considered: open loop (without measurement of the initial active force) and closed loop (with measurement-based feedback). In the open-loop case, neither activity nor odd mobility affects the average optimal work; however, the potential asymmetry enables the work extraction, and the oddness parameter κ shapes the protocol structure—introducing asymmetries and even backward jumps at the initial and final times. In the closed-loop setting, both activity and odd mobility shape the protocol design to enable work extraction. Optimal protocols exhibit directional asymmetries, including discontinuities and reversals, which depend on the alignment of the initial active force and the persistence time. Maximum work extraction occurs when the initial force aligns with the target at an intermediate persistence time, even in symmetric potentials. Furthermore, in regimes where such alignment disfavors work extraction for a normal tracer, odd mobility facilitates it. These findings highlight the potential for energy harvesting in active systems, driven by feedback and asymmetric interactions between different degrees of freedom.

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References (83)

  1. F. L. Curzon and B. Ahlborn, Efficiency of a carnot engine at maximum power output, Am. J. Phys. 43, 22 (1975).
  2. L. Chen, C. Wu, and F. Sun, Finite time thermodynamic optimization or entropy generation minimization of energy systems, J. Non-Equilib. Thermodyn. 24, 327 (1999).
  3. M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Efficiency at maximum power of low-dissipation carnot engines, Phys. Rev. Lett. 105, 150603 (2010).
  4. H. Wang and M. Pumera, Micro/nanomachines and living biosystems: From simple interactions to microcyborgs, Adv. Funct. Mater. 28, 1705421 (2018).
  5. B. Wang, Y. Zhang, and L. Zhang, Recent progress on micro- and nano-robots: Towards in vivo tracking and localization, Quant. Imaging Med. Surg. 8, 461 (2018).
  6. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  7. K. Goswami, Heat fluctuation of a harmonically trapped particle in an active bath, Phys. Rev. E 99, 012112 (2019).
  8. V. Holubec and R. Marathe, Underdamped active Brownian heat engine, Phys. Rev. E 102, 060101(R) (2020).
  9. K. Goswami, Work fluctuations in a generalized Gaussian active bath, Physica A 566, 125609 (2021).
  10. L. Cocconi and L. Chen, Efficiency of an autonomous, dynamic information engine operating on a single active particle, Phys. Rev. E 110, 014602 (2024).
  11. I. Derényi and R. D. Astumian, Efficiency of Brownian heat engines, Phys. Rev. E 59, R6219 (1999).
  12. I. Derényi, M. Bier, and R. D. Astumian, Generalized efficiency and its application to microscopic engines, Phys. Rev. Lett. 83, 903 (1999).
  13. G. E. Crooks, Measuring thermodynamic length, Phys. Rev. Lett. 99, 100602 (2007).
  14. S. Toyabe, T. Watanabe-Nakayama, T. Okamoto, S. Kudo, and E. Muneyuki, Thermodynamic efficiency and mechanochemical coupling of f1-atpase, Proc. Natl. Acad. Sci. USA 108, 17951 (2011).
  15. Y. Jun and P.-Y. Lai, Minimal dissipation protocols of an instantaneous equilibrium Brownian particle under time-dependent temperature and potential variations, Phys. Rev. Res. 4, 023157 (2022).
  16. T. Schmiedl and U. Seifert, Efficiency at maximum power: An analytically solvable model for stochastic heat engines, Europhys. Lett. 81, 20003 (2008).
  17. V. Blickle and C. Bechinger, Realization of a micrometre-sized stochastic heat engine, Nat. Phys. 8, 143 (2012).
  18. I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, Brownian carnot engine, Nat. Phys. 12, 67 (2016).
  19. V. Holubec and A. Ryabov, Cycling tames power fluctuations near optimum efficiency, Phys. Rev. Lett. 121, 120601 (2018).
  20. S. A. M. Loos, S. Monter, F. Ginot, and C. Bechinger, Universal symmetry of optimal control at the microscale, Phys. Rev. X 14, 021032 (2024).
  21. J. A. C. Albay, S. R. Wulaningrum, C. Kwon, P.-Y. Lai, and Y. Jun, Thermodynamic cost of a shortcuts-to-isothermal transport of a Brownian particle, Phys. Rev. Res. 1, 033122 (2019).
  22. J. A. C. Albay, C. Kwon, P.-Y. Lai, and Y. Jun, Work relation in instantaneous-equilibrium transition of forward and reverse processes, New J. Phys. 22, 123049 (2020).
  23. J. A. C. Albay, P.-Y. Lai, and Y. Jun, Realization of finite-rate isothermal compression and expansion using optical feedback trap, Appl. Phys. Lett. 116, 103706 (2020).
  24. Y. Jun and P.-Y. Lai, Instantaneous equilibrium transition for Brownian systems under time-dependent temperature and potential variations: Reversibility, heat and work relations, and fast isentropic process, Phys. Rev. Res. 3, 033130 (2021).
  25. A. Gomez-Marin, T. Schmiedl, and U. Seifert, Optimal protocols for minimal work processes in underdamped stochastic thermodynamics, J. Chem. Phys. 129, 024114 (2008).
  26. E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Optimal protocols and optimal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011).
  27. S. Blaber, M. D. Louwerse, and D. A. Sivak, Steps minimize dissipation in rapidly driven stochastic systems, Phys. Rev. E 104, L022101 (2021).
  28. D. A. Sivak and G. E. Crooks, Thermodynamic metrics and optimal paths, Phys. Rev. Lett. 108, 190602 (2012).
  29. S. Blaber and D. A. Sivak, Optimal control in stochastic thermodynamics, J. Phys. Commun. 7, 033001 (2023).
  30. M. C. Engel, J. A. Smith, and M. P. Brenner, Optimal control of nonequilibrium systems through automatic differentiation, Phys. Rev. X 13, 041032 (2023).
  31. C. A. Plata, D. Guéry-Odelin, E. Trizac, and A. Prados, Optimal work in a harmonic trap with bounded stiffness, Phys. Rev. E 99, 012140 (2019).
  32. S. Blaber and D. A. Sivak, Optimal control with a strong harmonic trap, Phys. Rev. E 106, L022103 (2022).
  33. A. Zhong and M. R. DeWeese, Limited-control optimal protocols arbitrarily far from equilibrium, Phys. Rev. E 106, 044135 (2022).
  34. P. Abiuso, V. Holubec, J. Anders, Z. Ye, F. Cerisola, and M. Perarnau-Llobet, Thermodynamics and optimal protocols of multidimensional quadratic Brownian systems, J. Phys. Commun. 6, 063001 (2022).
  35. L. Cocconi, J. Knight, and C. Roberts, Optimal power extraction from active particles with hidden states, Phys. Rev. Lett. 131, 188301 (2023).
  36. L. K. Davis, K. Proesmans, and É. Fodor, Active matter under control: Insights from response theory, Phys. Rev. X 14, 011012 (2024).
  37. T. Schmiedl and U. Seifert, Optimal finite-time processes in stochastic thermodynamics, Phys. Rev. Lett. 98, 108301 (2007).
  38. J. A. C. Albay, Z.-Y. Zhou, C.-H. Chang, H.-Y. Chen, J. S. Lee, C.-H. Chang, and Y. Jun, Colloidal heat engine driven by engineered active noise, Phys. Rev. Res. 5, 033208 (2023).
  39. Y. Wen, Z. Li, H. Wang, J. Zheng, J. Tang, P.-Y. Lai, X. Xu, and P. Tong, Activity-assisted barrier crossing of self-propelled colloids over parallel microgrooves, Phys. Rev. E 107, L032601 (2023).
  40. Y. Wen, Z. Li, H. Wang, J. Zheng, J. Tang, X. Xu, P.-Y. Lai, and P. Tong, Ratchet effect of self-propelled colloids in an asymmetric periodic potential, J. Chem. Phys. 162, 204903 (2025).
  41. D. Gupta, S. H. L. Klapp, and D. A. Sivak, Efficient control protocols for an active ornstein-uhlenbeck particle, Phys. Rev. E 108, 024117 (2023).
  42. 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).
  43. C. Casert and S. Whitelam, Learning protocols for the fast and efficient control of active matter, Nat. Commun. 15, 9128 (2024).
  44. 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).
  45. S. C. Takatori, T. Quah, and J. B. Rawlings, Feedback control of active matter, Annu. Rev. Condens. Matter Phys. 16, 319 (2025).
  46. J. Toner, Y. Tu, and S. Ramaswamy, Hydrodynamics and phases of flocks, Ann. Phys. (NY) 318, 170 (2005).
  47. N. C. Darnton, L. Turner, S. Rojevsky, and H. C. Berg, Dynamics of bacterial swarming, Biophys. J. 98, 2082 (2010).
  48. A. Zöttl and H. Stark, Emergent behavior in active colloids, J. Phys.: Condens. Matter 28, 253001 (2016).
  49. H. R. Vutukuri, M. Lisicki, E. Lauga, and J. Vermant, Light-switchable propulsion of active particles with reversible interactions, Nat. Commun. 11, 2628 (2020).
  50. R. Bebon, J. F. Robinson, and T. Speck, Thermodynamics of active matter: Tracking dissipation across scales, Phys. Rev. X 15, 021050 (2025).
  51. S. A. M. Loos and S. H. L. Klapp, Irreversibility, heat and information flows induced by non-reciprocal interactions, New J. Phys. 22, 123051 (2020).
  52. S. M. J. Khadem, R. Klages, and S. H. L. Klapp, Stochastic thermodynamics of fractional Brownian motion, Phys. Rev. Res. 4, 043186 (2022).
  53. H.-M. Chun, X. Durang, and J. D. Noh, Emergence of nonwhite noise in Langevin dynamics with magnetic Lorentz force, Phys. Rev. E 97, 032117 (2018).
  54. H. D. Vuijk, J. M. Brader, and A. Sharma, Anomalous fluxes in overdamped Brownian dynamics with lorentz force, J. Stat. Mech. (2019) 063203.
  55. J.-M. Park and H. Park, Thermodynamic uncertainty relation in the overdamped limit with a magnetic Lorentz force, Phys. Rev. Res. 3, 043005 (2021).
  56. E. Kalz, H. D. Vuijk, I. Abdoli, J.-U. Sommer, H. Löwen, and A. Sharma, Collisions enhance self-diffusion in odd-diffusive systems, Phys. Rev. Lett. 129, 090601 (2022).
  57. M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annu. Rev. Condens. Matter Phys. 14, 471 (2023).
  58. P. L. Muzzeddu, E. Kalz, A. Gambassi, A. Sharma, and R. Metzler, Self-diffusion anomalies of an odd tracer in soft-core media, New J. Phys. 27, 033025 (2025).
  59. F. Effenberger, H. Fichtner, K. Scherer, S. Barra, J. Kleimann, and R. D. Strauss, A generalized diffusion tensor for fully anisotropic diffusion of energetic particles in the heliospheric magnetic field, Astrophys. J. 750, 108 (2012).
  60. D. Schick, M. Weißenhofer, L. Rózsa, J. Rothörl, P. Virnau, and U. Nowak, Two levels of topology in skyrmion lattice dynamics, Phys. Rev. Res. 6, 013097 (2024).
  61. A. P. Petroff, X.-L. Wu, and A. Libchaber, Fast-moving bacteria self-organize into active two-dimensional crystals of rotating cells, Phys. Rev. Lett. 114, 158102 (2015).
  62. C. W. Chan, D. Wu, K. Qiao, K. L. Fong, Z. Yang, Y. Han, and R. Zhang, Chiral active particles are sensitive reporters to environmental geometry, Nat. Commun. 15, 1406 (2024).
  63. J.-L. Auriault, C. Moyne, and H. P. A. Souto, On the asymmetry of the dispersion tensor in porous media, Transp. Porous Media 85, 771 (2010).
  64. C. J. O. Reichhardt and C. Reichhardt, Active rheology in odd-viscosity systems, Europhys. Lett. 137, 66004 (2022).
  65. X. Cao, D. Das, N. Windbacher, F. Ginot, M. Krüger, and C. Bechinger, Memory-induced magnus effect, Nat. Phys. 19, 1904 (2023).
  66. A. Shalchi, Applicability of the Taylor-Green-Kubo formula in particle diffusion theory, Phys. Rev. E 83, 046402 (2011).
  67. K. Goswami and R. Metzler, Effects of active noise on transition-path dynamics, J. Phys. Complexity 4, 025005 (2023).
  68. K. Goswami and K. L. Sebastian, Diffusion caused by two noises—active and thermal, J. Stat. Mech. (2019) 083501.
  69. C. Maggi, M. Paoluzzi, N. Pellicciotta, A. Lepore, L. Angelani, and R. Di Leonardo, Generalized energy equipartition in harmonic oscillators driven by active baths, Phys. Rev. Lett. 113, 238303 (2014).
  70. 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).
  71. K. Sekimoto, Langevin equation and thermodynamics, Prog. Theor. Phys. Suppl. 130, 17 (1998).
  72. L. Caprini, U. M. B. Marconi, A. Puglisi, and A. Vulpiani, The entropy production of ornstein–uhlenbeck active particles: A path integral method for correlations, J. Stat. Mech. (2019) 053203.
  73. D. Martin, J. O’Byrne, M. E. Cates, É. Fodor, C. Nardini, J. Tailleur, and F. van Wijland, Statistical mechanics of active ornstein-uhlenbeck particles, Phys. Rev. E 103, 032607 (2021).
  74. K. Goswami, Inertial particle under active fluctuations: Diffusion and work distributions, Phys. Rev. E 105, 044123 (2022).
  75. K. Goswami, A. G. Cherstvy, A. Godec, and R. Metzler, Anomalous diffusion of active Brownian particles in responsive elastic gels: Nonergodicity, non-Gaussianity, and distributions of trapping times, Phys. Rev. E 110, 044609 (2024).
  76. See Supplemental Material at http://link.aps.org/supplemental/10.1103/qrvl-mng6 for additional figures.
  77. L. Caprini, H. Löwen, and U. Marini Bettolo Marconi, Chiral active matter in external potentials, Soft Matter 19, 6234 (2023).
  78. W. R. DiLuzio, L. Turner, M. Mayer, P. Garstecki, D. B. Weibel, H. C. Berg, and G. M. Whitesides, Escherichia coli swim on the right-hand side, Nature (London) 435, 1271 (2005).
  79. F. V. Reyes, M. A. López-Castaño, and Á. Rodríguez-Rivas, Diffusive regimes in a two-dimensional chiral fluid, Commun. Phys. 5, 256 (2022).
  80. C. Hargus, F. Ghimenti, J. Tailleur, and F. van Wijland, Odd dynamics of passive objects in a chiral active bath, Phys. Rev. Lett. 135, 167102 (2025).
  81. K. Goswami, Dataset for “Optimal work control for an odd-diffusive tracer under active fluctuations”, Zenodo (2025), http://doi.org/10.5281/zenodo.16808964.
  82. C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 2004), Vol. 3.
  83. I. Gelfand and S. Fomin, Calculus of Variations (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963).

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Phys. Rev. Research 7, 043309– Published 17 December, 2025
  • Received 18 August 2025
  • Accepted 23 November 2025
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