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Optimal control of a mesoscopic information engine

Emanuele Panizon

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Phys. Rev. E 114, 034144 – Published 28 September, 2026

DOI: https://doi.org/10.1103/85qf-nwj5

Abstract

We analytically solve the finite-time control problem of driving an overdamped particle via an optical trap under costly measurement. By formulating this mesoscopic information engine within the partially observable Markov decision process framework, we demonstrate that the underlying linear-quadratic-Gaussian dynamics decouples the optimal measurement and driving protocols. We derive the optimal feedback control law for the trap placement, which recovers the discontinuous Schmiedl-Seifert driving protocol in the open-loop limit and extends it to any measurement scheduling. For a costly binary (on-off) sensor, we evaluate the optimal measurement protocol and derive physical bounds on the maximum net gain that can be extracted from thermal fluctuations. We show the emergence of deadline-induced blindness, a phenomenon where all measurements cease as the deadline approaches regardless of their cost. Taking the infinite-horizon limit, we find the exact periodic measurement schedules for the steady state as a function of the measurement cost C and derive the macroscopic velocity envelopes beyond which viscous drag forces the engine into a net-dissipative regime. Finally, we generalize the results to a variable-precision sensor.

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References (52)

  1. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  2. G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721 (1999).
  3. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  4. T. Sagawa and M. Ueda, Generalized Jarzynski equality under nonequilibrium feedback control, Phys. Rev. Lett. 104, 090602 (2010).
  5. M. Bauer, D. Abreu, and U. Seifert, Efficiency of a Brownian information machine, J. Phys. A: Math. Theor. 45, 162001 (2012).
  6. T. Sagawa and M. Ueda, Nonequilibrium thermodynamics of feedback control, Phys. Rev. E 85, 021104 (2012).
  7. J. Ehrich and D. A. Sivak, Energy and information flows in autonomous systems, Front. Phys. 11, 1108357 (2023).
  8. J. C. Maxwell, Theory of Heat (Longmans, Green, and Co., London, 1871).
  9. L. Szilard, öber die entropieverminderung in einem thermodynamischen system bei eingriffen intelligenter wesen, Z. Phys. 53, 840 (1929).
  10. R. Landauer, Irreversibility and heat generation in the computing process, IBM J. Res. Dev. 5, 183 (1961).
  11. J. M. Parrondo, J. M. Horowitz, and T. Sagawa, Thermodynamics of information, Nat. Phys. 11, 131 (2015).
  12. S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality, Nat. Phys. 6, 988 (2010).
  13. A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental verification of Landauer's principle linking information and thermodynamics, Nature (London) 483, 187 (2012).
  14. I. A. Martínez, É. Roldán, L. Dinis, and R. A. Rica, Colloidal heat engines: A review, Soft Matter 13, 22 (2017).
  15. G. Paneru and H. Kyu Pak, Colloidal engines for innovative tests of information thermodynamics, Adv. Phys. X 5, 1823880 (2020).
  16. T. K. Saha, J. N. Lucero, J. Ehrich, D. A. Sivak, and J. Bechhoefer, Maximizing power and velocity of an information engine, Proc. Natl. Acad. Sci. USA 118, e2023356118 (2021).
  17. T. K. Saha, J. Ehrich, M. Gavrilov, S. Still, D. A. Sivak, and J. Bechhoefer, Information engine in a nonequilibrium bath, Phys. Rev. Lett. 131, 057101 (2023).
  18. A. Bensoussan, Stochastic Control of Partially Observable Systems (Cambridge University Press, Cambridge, 1992).
  19. J. Alvarado, E. G. Teich, D. A. Sivak, and J. Bechhoefer, Optimal control in soft and active matter, Annu. Rev. Condens. Matter Phys. 17, 327 (2026).
  20. E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Optimal protocols and optimal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011).
  21. T. Kamijima, A. Takatsu, K. Funo, and T. Sagawa, Optimal finite-time Maxwell's demons in Langevin systems, Phys. Rev. Res. 7, 023159 (2025).
  22. A. T. Mohite and H. Rieger, Generalized finite-time optimal control framework in stochastic thermodynamics, arXiv:2511.00974.
  23. S. Blaber, M. D. Louwerse, and D. A. Sivak, Steps minimize dissipation in rapidly driven stochastic systems, Phys. Rev. E 104, L022101 (2021).
  24. T. Schmiedl and U. Seifert, Optimal finite-time processes in stochastic thermodynamics, Phys. Rev. Lett. 98, 108301 (2007).
  25. D. Abreu and U. Seifert, Extracting work from a single heat bath through feedback, Europhys. Lett. 94, 10001 (2011).
  26. 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).
  27. 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).
  28. S. Whitelam, Demon in the machine: Learning to extract work and absorb entropy from fluctuating nanosystems, Phys. Rev. X 13, 021005 (2023).
  29. J. Bechhoefer, Hidden Markov models for stochastic thermodynamics, New J. Phys. 17, 075003 (2015).
  30. M. Biehl and N. Virgo, Communications in Computer and Information Science, in Active Inference, edited by C. L. Buckley, D. Cialfi, P. Lanillos, M. Ramstead, N. Sajid, H. Shimazaki, and T. Verbelen (Springer, Cham, 2022), Vol. 1721, pp. 16–31.
  31. L. Meier, J. Peschon, and R. Dressler, Optimal control of measurement subsystems, IEEE Trans. Autom. Control 12, 528 (1967).
  32. B. D. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods (Courier, North Chelmsford, 2007).
  33. J. Bechhoefer, Control Theory for Physicists (Cambridge University Press, Cambridge, 2021).
  34. K. J. Åström, Optimal control of Markov processes with incomplete state information, J. Math. Anal. Appl. 10, 174 (1965).
  35. L. P. Kaelbling, M. L. Littman, and A. R. Cassandra, Planning and acting in partially observable stochastic domains, Artif. Intell. 101, 99 (1998).
  36. J. M. Horowitz and M. Esposito, Thermodynamics with continuous information flow, Phys. Rev. X 4, 031015 (2014).
  37. G. Falasco and M. Esposito, Macroscopic stochastic thermodynamics, Rev. Mod. Phys. 97, 015002 (2025).
  38. J. V. Koski, A. Kutvonen, I. M. Khaymovich, T. Ala-Nissila, and J. P. Pekola, On-chip Maxwell's demon as an information-powered refrigerator, Phys. Rev. Lett. 115, 260602 (2015).
  39. G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev. 36, 823 (1930).
  40. D. T. Gillespie, Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral, Phys. Rev. E 54, 2084 (1996).
  41. R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957).
  42. D. Bertsekas, Dynamic Programming and Optimal Control, 4th ed. (Athena Scientific, Nashua, 2012), Vol. 1.
  43. H. A. Simon, Dynamic programming under uncertainty with a quadratic criterion function, Econometrica 24, 74 (1956).
  44. Y. Bar-Shalom and E. Tse, Dual effect, certainty equivalence, and separation in stochastic control, IEEE Trans. Autom. Control 19, 494 (1974).
  45. T. Sagawa and M. Ueda, Minimal energy cost for thermodynamic information processing: Measurement and information erasure, Phys. Rev. Lett. 102, 250602 (2009).
  46. R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W function, Adv. Comput. Math. 5, 329 (1996).
  47. R. Kalman, A new approach to linear filtering and prediction problems, J. Basic Eng. 82, 35 (1960).
  48. L. L. Bonilla, Active Ornstein-Uhlenbeck particles, Phys. Rev. E 100, 022601 (2019).
  49. L. K. Davis, Optimal multi-parameter control of trapped active matter, arXiv:2603.16778.
  50. L. Cocconi, J. Knight, and C. Roberts, Optimal power extraction from active particles with hidden states, Phys. Rev. Lett. 131, 188301 (2023).
  51. C. Casert and S. Whitelam, Learning protocols for the fast and efficient control of active matter, Nat. Commun. 15, 9128 (2024).
  52. E. Panizon, RitAreaSciencePark/information-engine: v1.2.0: Rebuttal manuscript submission, Zenodo, 2026, https://doi.org/10.5281/zenodo.21774633.

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Phys. Rev. E 114, 034144– Published 28 September, 2026
  • Received 10 April 2026
  • Accepted 7 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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