- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Generalized finite-time optimal control framework in stochastic thermodynamics

Atul Tanaji Mohite* and Heiko Rieger

  • Department of Theoretical Physics and Center for Biophysics, Saarland University, Saarbrücken, Germany

  • *Contact author: atul.mohite@uni-saarland.de
PDF

Phys. Rev. Research 8, 033340 – Published 21 September, 2026

DOI: https://doi.org/10.1103/bc7b-tfl6

Abstract

Optimal processes in stochastic thermodynamics represent a frontier for understanding the control and design of nonequilibrium systems, with broad practical applications in biology, chemistry, and nanoscale/mesoscale systems. Optimal transport theory and thermodynamic geometry have emerged as leading optimal control methodologies, but both rely on slow-driving and close-to-equilibrium assumptions. An optimal control framework in stochastic thermodynamics for finite-time driving remains elusive. Here, we solve an optimal control problem for driving the control parameters of a discrete-state far-from-equilibrium process from an initial to a final value in finite time. Optimal driving protocols are derived that minimize the total finite-time dissipation cost of the driving process. Our framework reveals that discontinuous end-point jumps are a generic, model-independent physical mechanism that minimizes the optimal driving entropy production—“geometric thermodynamic far-from-equilibrium shortcuts in swift state-to-state transformations”—whose importance is further amplified in far-from-equilibrium systems. The thermodynamic and dynamical interpretation of discontinuous end-point jumps is formulated. An exact mapping between the finite-time and slow-driving optimal control formulations is elucidated, advancing the state of the art in optimal transport theory and thermodynamic geometry, which has been the prevailing paradigm for studying optimal processes in stochastic thermodynamics under slow-driving assumptions. Our framework opens up a broad range of applications to the thermodynamically efficient control of far-from-equilibrium systems in finite time, and thereby a route toward their efficient design principles.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (107)

  1. N. Shiraishi, An Introduction to Stochastic Thermodynamics: From Basic to Advanced (Springer Nature Singapore, Singapore, 2023), pp. 31–47.
  2. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  3. K. Sekimoto, Stochastic Energetics (Springer, Berlin, Heidelberg, 2010).
  4. S. Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Phys. Rev. X 7, 021051 (2017).
  5. A. T. Mohite and H. Rieger, Stochastic thermodynamics of nonreciprocally interacting particles and fields, Phys. Rev. E 113, 064136 (2026).
  6. A. T. Mohite and H. Rieger, Thermodynamically consistent coarse-graining: From interacting particles to fields via second quantization, arXiv:2508.11430.
  7. A. T. Mohite and H. Rieger, Minimum action principle for entropy production rate of far-from-equilibrium systems, arXiv:2511.00967.
  8. A. T. Mohite and H. Rieger, Thermodynamic length in stochastic thermodynamics of far-from-equilibrium systems: Unification of fluctuation relation and thermodynamic uncertainty relation, Phys. Rev. E 114, 014103 (2026).
  9. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  10. T. Schmiedl and U. Seifert, Optimal finite-time processes in stochastic thermodynamics, Phys. Rev. Lett. 98, 108301 (2007).
  11. A. Zhong and M. R. DeWeese, Beyond linear response: Equivalence between thermodynamic geometry and optimal transport, Phys. Rev. Lett. 133, 057102 (2024).
  12. 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).
  13. These different methodologies are related to different optimal control formulations of classical stochastic systems discussed subsequently, but they are not necessarily the same. Fundamental differences between them can be drastic, despite their case-specific applicability. First, their specific aims differ significantly depending on what they are designed for, mainly the dynamical aspects (see Ref. [12]). Second, the nonequilibrium notion of entropy production rate itself is not necessarily well defined for quantum thermodynamic systems, but it is of paramount importance for classical stochastic thermodynamic systems. As a result, these quantum methodologies strictly focus on the optimization of dynamical aspects, such as constraints on the dynamical evolution of the quantum state or operators, or on their eigenstates. These methods justify the importance of dynamical aspects over thermodynamic aspects for finite-time optimal processes, arguing that dynamics become more significant than thermodynamics. However, by construction, the most fundamental principles of statistical physics imply that thermodynamics constrains the dynamics, and not the other way around. Therefore, even the dynamics of finite-time optimal processes must be constrained by thermodynamics. We rigorously prove this within our framework.
  14. A. Gomez-Marin, T. Schmiedl, and U. Seifert, Optimal protocols for minimal work processes in underdamped stochastic thermodynamics, J. Chem. Phys. 129, 024114 (2008).
  15. H. Then and A. Engel, Computing the optimal protocol for finite-time processes in stochastic thermodynamics, Phys. Rev. E 77, 041105 (2008).
  16. M. V. S. Bonança and S. Deffner, Minimal dissipation in processes far from equilibrium, Phys. Rev. E 98, 042103 (2018).
  17. M. C. Engel, J. A. Smith, and M. P. Brenner, Optimal control of nonequilibrium systems through automatic differentiation, Phys. Rev. X 13, 041032 (2023).
  18. C. R. Rao et al., Information and the accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc 37, 81 (1945).
  19. F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem. Phys. 63, 2479 (1975).
  20. G. Ruppeiner, Riemannian geometry in thermodynamic fluctuation theory, Rev. Mod. Phys. 67, 605 (1995).
  21. P. Salamon, J. D. Nulton, and R. S. Berry, Length in statistical thermodynamics, J. Chem. Phys. 82, 2433 (1985).
  22. F. Schlögl, Thermodynamic metric and stochastic measures, Z. Phys. B 59, 449 (1985).
  23. D. Brody and N. Rivier, Geometrical aspects of statistical mechanics, Phys. Rev. E 51, 1006 (1995).
  24. G. E. Crooks, Measuring thermodynamic length, Phys. Rev. Lett. 99, 100602 (2007).
  25. K. Sekimoto and S.-I. Sasa, Complementarity relation for irreversible process derived from stochastic energetics, J. Phys. Soc. Jpn. 66, 3326 (1997).
  26. D. A. Sivak and G. E. Crooks, Thermodynamic metrics and optimal paths, Phys. Rev. Lett. 108, 190602 (2012).
  27. D. Mandal and C. Jarzynski, Analysis of slow transitions between nonequilibrium steady states, J. Stat. Mech. (2016) 063204.
  28. D. A. Sivak and G. E. Crooks, Thermodynamic geometry of minimum-dissipation driven barrier crossing, Phys. Rev. E 94, 052106 (2016).
  29. P. R. Zulkowski, D. A. Sivak, G. E. Crooks, and M. R. DeWeese, Geometry of thermodynamic control, Phys. Rev. E 86, 041148 (2012).
  30. P. R. Zulkowski, D. A. Sivak, and M. R. DeWeese, Optimal control of transitions between nonequilibrium steady states, PLoS One 8, e82754 (2013).
  31. P. R. Zulkowski and M. R. DeWeese, Optimal finite-time erasure of a classical bit, Phys. Rev. E 89, 052140 (2014).
  32. P. R. Zulkowski and M. R. DeWeese, Optimal protocols for slowly driven quantum systems, Phys. Rev. E 92, 032113 (2015).
  33. G. M. Rotskoff, G. E. Crooks, and E. Vanden-Eijnden, Geometric approach to optimal nonequilibrium control: Minimizing dissipation in nanomagnetic spin systems, Phys. Rev. E 95, 012148 (2017).
  34. D. Loutchko, Y. Sughiyama, and T. J. Kobayashi, Riemannian geometry of optimal driving and thermodynamic length and its application to chemical reaction networks, Phys. Rev. Res. 4, 043049 (2022).
  35. J.-F. Chen, C. P. Sun, and H. Dong, Extrapolating the thermodynamic length with finite-time measurements, Phys. Rev. E 104, 034117 (2021).
  36. G. Li, J.-F. Chen, C. P. Sun, and H. Dong, Geodesic path for the minimal energy cost in shortcuts to isothermality, Phys. Rev. Lett. 128, 230603 (2022).
  37. G. Watanabe and Y. Minami, Finite-time thermodynamics of fluctuations in microscopic heat engines, Phys. Rev. Res. 4, L012008 (2022).
  38. S. Ito and A. Dechant, Stochastic time evolution, information geometry, and the Cramér-Rao bound, Phys. Rev. X 10, 021056 (2020).
  39. Y.-H. Ma, R.-X. Zhai, J. Chen, C. P. Sun, and H. Dong, Experimental test of the 1/τ-scaling entropy generation in finite-time thermodynamics, Phys. Rev. Lett. 125, 210601 (2020).
  40. S.-I. Amari, Information Geometry and Its Applications, Applied Mathematical Sciences (Springer, Tokyo, 2016).
  41. R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal. 29, 1 (1998).
  42. J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math. 84, 375 (2000).
  43. C. Villani, Topics in Optimal Transportation (American Mathematical Society, Providence, 2003).
  44. F. Santambrogio, Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications (Birkhäuser, Cham, 2015), Vol. 87.
  45. E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Optimal protocols and optimal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011).
  46. E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Boundary layers in stochastic thermodynamics, Phys. Rev. E 85, 020103(R) (2012).
  47. E. Aurell, K. Gawedzki, C. Mejía-Monasterio, R. Mohayaee, and P. Muratore-Ginanneschi, Refined second law of thermodynamics for fast random processes, J. Stat. Phys. 147, 487 (2012).
  48. Y. Chen, T. T. Georgiou, and A. Tannenbaum, Stochastic control and nonequilibrium thermodynamics: Fundamental limits, IEEE Trans. Autom. Control 65, 2979 (2019).
  49. Y. Zhang, Work needed to drive a thermodynamic system between two distributions, Europhys. Lett. 128, 30002 (2020).
  50. Y. Zhang, Optimization of stochastic thermodynamic machines, J. Stat. Phys. 178, 1336 (2020).
  51. A. Dechant and Y. Sakurai, Thermodynamic interpretation of Wasserstein distance, arXiv:1912.08405.
  52. M. Nakazato and S. Ito, Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance, Phys. Rev. Res. 3, 043093 (2021).
  53. R. Fu, A. Taghvaei, Y. Chen, and T. T. Georgiou, Maximal power output of a stochastic thermodynamic engine, Automatica 123, 109366 (2021).
  54. A. Taghvaei, O. M. Miangolarra, R. Fu, Y. Chen, and T. T. Georgiou, On the relation between information and power in stochastic thermodynamic engines, IEEE Control Syst. Lett. 6, 434 (2022).
  55. T. Van Vu and K. Saito, Thermodynamic unification of optimal transport: Thermodynamic uncertainty relation, minimum dissipation, and thermodynamic speed limits, Phys. Rev. X 13, 011013 (2023).
  56. S. Ito, Geometric thermodynamics for the Fokker–Planck equation: Stochastic thermodynamic links between information geometry and optimal transport, Inf. Geom. 7, 441 (2024).
  57. R. Sabbagh, O. M. Miangolarra, and T. T. Georgiou, Wasserstein speed limits for Langevin systems, Phys. Rev. Res. 6, 033308 (2024).
  58. J. Klinger and G. M. Rotskoff, Universal energy-speed-accuracy trade-offs in driven nonequilibrium systems, Phys. Rev. E 111, 014114 (2025).
  59. S. Kolouri, S. R. Park, M. Thorpe, D. Slepcev, and G. K. Rohde, Optimal mass transport: Signal processing and machine-learning applications, IEEE Signal Process. Mag. 34, 43 (2017).
  60. S. Oikawa, Y. Nakayama, S. Ito, T. Sagawa, and S. Toyabe, Experimentally achieving minimal dissipation via thermodynamically optimal transport, Nat. Commun. 16, 10424 (2025).
  61. J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571 (1976).
  62. This threefold orthogonal decomposition of the mean dissipation was highlighted in Ref. [8] and generalized to a fourfold orthogonal decomposition in Ref. [5] for “actio = reactio” symmetry-breaking nonreciprocal systems.
  63. W. E, W. Ren, and E. Vanden-Eijnden, Minimum action method for the study of rare events, Commun. Pure Appl. Math. 57, 637 (2004).
  64. M. Heymann and E. Vanden-Eijnden, Pathways of maximum likelihood for rare events in nonequilibrium systems: Application to nucleation in the presence of shear, Phys. Rev. Lett. 100, 140601 (2008).
  65. E. Vanden-Eijnden and M. Heymann, The geometric minimum action method for computing minimum energy paths, J. Chem. Phys. 128, 061103 (2008).
  66. T. Grafke, T. Schäfer, and E. Vanden-Eijnden, Long term effects of small random perturbations on dynamical systems: Theoretical and computational tools, in Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, edited by R. Melnik, R. Makarov, and J. Belair (Springer, New York, 2017), pp. 17–55.
  67. T. Grafke and E. Vanden-Eijnden, Numerical computation of rare events via large deviation theory, Chaos 29, 063118 (2019).
  68. P. Gagrani and E. Smith, Action functional gradient descent algorithm for estimating escape paths in stochastic chemical reaction networks, Phys. Rev. E 107, 034305 (2023).
  69. R. Zakine and E. Vanden-Eijnden, Minimum-action method for nonequilibrium phase transitions, Phys. Rev. X 13, 041044 (2023).
  70. E. Smith, H. B. Smith, and J. L. Andersen, Rules, hypergraphs, and probabilities: The three-level analysis of chemical reaction systems and other stochastic stoichiometric population processes, PLoS Complex Syst. 1, e0000022 (2024).
  71. A. Cherukuri, B. Gharesifard, and J. Cortés, Saddle-point dynamics: Conditions for asymptotic stability of saddle points, SIAM J. Control Optim. 55, 486 (2017).
  72. T. Lin, C. Jin, and M. Jordan, On gradient descent ascent for nonconvex-concave minimax problems, in Proceedings of the 37th International Conference on Machine Learning (PMLR, 2020), Vol. 119, pp. 6083–6093.
  73. J. Yan, H. Touchette, and G. M. Rotskoff, Learning nonequilibrium control forces to characterize dynamical phase transitions, Phys. Rev. E 105, 024115 (2022).
  74. D.-K. Kim, Y. Bae, S. Lee, and H. Jeong, Learning entropy production via neural networks, Phys. Rev. Lett. 125, 140604 (2020).
  75. S. Otsubo, S. Ito, A. Dechant, and T. Sagawa, Estimating entropy production by machine learning of short-time fluctuating currents, Phys. Rev. E 101, 062106 (2020).
  76. S. K. Manikandan, D. Gupta, and S. Krishnamurthy, Inferring entropy production from short experiments, Phys. Rev. Lett. 124, 120603 (2020).
  77. S. Otsubo, S. K. Manikandan, T. Sagawa, and S. Krishnamurthy, Estimating time-dependent entropy production from non-equilibrium trajectories, Commun. Phys. 5, 11 (2022).
  78. S. A. Horiguchi and T. J. Kobayashi, Optimal control of stochastic reaction networks with entropic control cost and emergence of mode-switching strategies, PRX Life 3, 033027 (2025).
  79. T. Tottori and T. J. Kobayashi, Theory for optimal estimation and control under resource limitations and its applications to biological information processing and decision-making, Phys. Rev. Res. 7, 043048 (2025).
  80. H. J. Kappen, Linear theory for control of nonlinear stochastic systems, Phys. Rev. Lett. 95, 200201 (2005).
  81. H. J. Kappen, Path integrals and symmetry breaking for optimal control theory, J. Stat. Mech. (2005) P11011.
  82. The driving time τ is different from the previously used observation time τobs used to delineate the short-time and long-time scaling symmetries of 〈Σ̇〉.
  83. As highlighted in Sec. 1b, ψ̇E and 〈Σ̇ex〉 are boundary terms in time; therefore, their optimization leads to a trivial solution. To address this, the optimization problem must be defined for ψE and 〈Σex〉. In contrast, 〈Σ̇hk〉 cannot be integrated out in time; therefore, the optimization problem must be defined for 〈Σ̇hk〉.
  84. Optimal transport theory introduces an incremental development regarding (S3) that addresses (S2); however, it remains a fundamental shortcoming for fEQ systems (see Refs. [7, 8]). Therefore, by construction, methodologies with (S2) are also prone to (S3).
  85. By construction, methodologies with (S4) are also prone to (S5).
  86. There is a minor technical difference that is not of fundamental importance, since the mass defined for a control parameter is the inverse of the mass defined for its dual conjugate (see Sec.  5c).
  87. S. Blaber, M. D. Louwerse, and D. A. Sivak, Steps minimize dissipation in rapidly driven stochastic systems, Phys. Rev. E 104, L022101 (2021).
  88. S. Blaber and D. A. Sivak, Optimal control in stochastic thermodynamics, J. Phys. Commun. 7, 033001 (2023).
  89. H. Qian, Entropy production and excess entropy in a nonequilibrium steady-state of single macromolecules, Phys. Rev. E 65, 021111 (2002).
  90. S. Ito, Stochastic thermodynamic interpretation of information geometry, Phys. Rev. Lett. 121, 030605 (2018).
  91. P. B. Melo, S. M. Duarte Queirós, and W. A. M. Morgado, Stochastic thermodynamics of Fisher information, Phys. Rev. E 111, 014101 (2025).
  92. P. B. Melo, F. Iemini, D. O. Soares-Pinto, S. M. D. Queirós, and W. A. M. Morgado, Thermodynamic interpretation to stochastic Fisher information and single-trajectory speed limits, Phys. Rev. E 112, 014126 (2025).
  93. P. B. Melo, S. M. Duarte Queirós, D. O. Soares-Pinto, and W. A. M. Morgado, Classical geometric fluctuation relations, J. Phys. A: Math. Theor. 58, 195001 (2025).
  94. Note that we use the negative of the excess EP to ensure the positivity of the mass.
  95. L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. 91, 1505 (1953).
  96. S. Machlup and L. Onsager, Fluctuations and irreversible process. II. Systems with kinetic energy, Phys. Rev. 91, 1512 (1953).
  97. N. Shiraishi, K. Funo, and K. Saito, Speed limit for classical stochastic processes, Phys. Rev. Lett. 121, 070601 (2018).
  98. K. Yoshimura and S. Ito, Thermodynamic uncertainty relation and thermodynamic speed limit in deterministic chemical reaction networks, Phys. Rev. Lett. 127, 160601 (2021).
  99. S. Liepelt and R. Lipowsky, Kinesin’s network of chemomechanical motor cycles, Phys. Rev. Lett. 98, 258102 (2007).
  100. J. Ninio, Kinetic amplification of enzyme discrimination, Biochimie 57, 587 (1975).
  101. D. Sigg, Modeling ion channels: Past, present, and future, J. Gen. Physiol. 144, 7 (2014).
  102. J. J. Hopfield, Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity, Proc. Natl. Acad. Sci. USA 71, 4135 (1974).
  103. A. Murugan, D. A. Huse, and S. Leibler, Speed, accuracy, and dissipation in kinetic proofreading, Proc. Natl. Acad. Sci. USA 109, 12034 (2012).
  104. Note that the state-density conservation constraint should be used for the CRN instead of the state-space probability conservation constraint (∑{i}ρi=1) used here for the MJP.
  105. M. Okuyama and M. Ohzeki, Quantum speed limit is not quantum, Phys. Rev. Lett. 120, 070402 (2018).
  106. B. Shanahan, A. Chenu, N. Margolus, and A. del Campo, Quantum speed limits across the quantum-to-classical transition, Phys. Rev. Lett. 120, 070401 (2018).
  107. S. B. Nicholson, L. P. García-Pintos, A. del Campo, and J. R. Green, Time–information uncertainty relations in thermodynamics, Nat. Phys. 16, 1211 (2020).

Outline

Information

Phys. Rev. Research 8, 033340– Published 21 September, 2026
  • Received 12 January 2026
  • Accepted 6 August 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