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  • Open Access

Identifying nonequilibrium degrees of freedom in high-dimensional stochastic systems

Catherine Ji1,2,*, Ravin Raj1,*, Benjamin Eysenbach2,†, and Gautam Reddy1,‡

  • *These authors contributed equally to this work.
  • †Contact author: eysenbach@princeton.edu
  • ‡Contact author: greddy@princeton.edu
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Phys. Rev. Research 8, 023264 – Published 8 June, 2026

DOI: https://doi.org/10.1103/1ncx-6cxk

Abstract

Any coarse-grained description of a nonequilibrium system should faithfully represent its latent irreversible degrees of freedom. However, standard dimensionality reduction methods typically prioritize accurate reconstruction over physical relevance. Here, we introduce a model-free approach to identify irreversible degrees of freedom in stochastic systems that are in a nonequilibrium steady state. Our method leverages the insight that a black-box classifier, trained to differentiate between forward and time-reversed trajectories, implicitly estimates the local entropy production rate. By parametrizing this classifier as a quadratic form of learned state representations, we obtain nonlinear embeddings of high-dimensional state-space dynamics, which we term latent embeddings of nonequilibrium systems (LENS). LENS effectively identifies low-dimensional irreversible flows and provides a scalable, learning-based strategy for estimating entropy production rates directly from high-dimensional time series data.

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References (42)

  1. C. Battle, C. P. Broedersz, N. Fakhri, V. F. Geyer, J. Howard, C. F. Schmidt, and F. C. MacKintosh, Broken detailed balance at mesoscopic scales in active biological systems, Science 352, 604 (2016).
  2. J. Gladrow, C. P. Broedersz, and C. F. Schmidt, Nonequilibrium dynamics of probe filaments in actin-myosin networks, Phys. Rev. E 96, 022408 (2017).
  3. C. W. Lynn, C. M. Holmes, W. Bialek, and D. J. Schwab, Emergence of local irreversibility in complex interacting systems, Phys. Rev. E 106, 034102 (2022).
  4. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  5. H. Qian, S. Kjelstrup, A. B. Kolomeisky, and D. Bedeaux, Entropy production in mesoscopic stochastic thermodynamics: Nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces, J. Phys.: Condens. Matter 28, 153004 (2016).
  6. F. S. Gnesotto, F. Mura, J. Gladrow, and C. P. Broedersz, Broken detailed balance and non-equilibrium dynamics in living systems: A review, Rep. Prog. Phys. 81, 066601 (2018).
  7. F. Mura, G. Gradziuk, and C. P. Broedersz, Nonequilibrium scaling behavior in driven soft biological assemblies, Phys. Rev. Lett. 121, 038002 (2018).
  8. T. H. Tan, G. A. Watson, Y.-C. Chao, J. Li, T. R. Gingrich, J. M. Horowitz, and N. Fakhri, Scale-dependent irreversibility in living matter, arXiv:2107.05701.
  9. Q. Wang, S. Kulkarni, and S. Verdu, Divergence estimation of continuous distributions based on data-dependent partitions, IEEE Trans. Inf. Theory 51, 3064 (2005).
  10. J. Ziv and N. Merhav, A measure of relative entropy between individual sequences with application to universal classification, IEEE Trans. Inf. Theory 39, 1270 (1993).
  11. R. Roldán and J. M. R. Parrondo, Entropy production and Kullback-Leibler divergence between stationary trajectories of discrete systems, Phys. Rev. E 85, 031129 (2012).
  12. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  13. T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluctuations, Phys. Rev. Lett. 116, 120601 (2016).
  14. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  15. J. Li, J. M. Horowitz, T. R. Gingrich, and N. Fakhri, Quantifying dissipation using fluctuating currents, Nat. Commun. 10, 1666 (2019).
  16. E. Roldán, J. Barral, P. Martin, J. M. R. Parrondo, and F. Jülicher, Quantifying entropy production in active fluctuations of the hair-cell bundle from time irreversibility and uncertainty relations, New J. Phys. 23, 083013 (2021).
  17. F. S. Gnesotto, G. Gradziuk, P. Ronceray, and C. P. Broedersz, Learning the non-equilibrium dynamics of Brownian movies, Nat. Commun. 11, 5378 (2020).
  18. D.-K. Kim, Y. Bae, S. Lee, and H. Jeong, Learning entropy production via neural networks, Phys. Rev. Lett. 125, 140604 (2020).
  19. Y. Bae, D.-K. Kim, and H. Jeong, Inferring dissipation maps from videos using convolutional neural networks, Phys. Rev. Res. 4, 033094 (2022).
  20. J. Lyu, K. J. Ray, and J. P. Crutchfield, Learning entropy production from underdamped Langevin trajectories, Phys. Rev. E 110, 064151 (2024).
  21. M. Vodret, C. Pacini, and C. Bongiorno, Functional decomposition and estimation of irreversibility in time series via machine learning, Phys. Rev. E 110, 064310 (2024).
  22. N. M. Boffi and E. Vanden-Eijnden, Model-free learning of probability flows: Elucidating the nonequilibrium dynamics of flocking, Phys. Rev. Lett. 135, 238301 (2025) .
  23. N. M. Boffi and E. Vanden-Eijnden, Deep learning probability flows and entropy production rates in active matter, Proc. Natl. Acad. Sci. USA 121, e2318106121 (2024).
  24. M. S. Schmitt, M. Koch-Janusz, M. Fruchart, D. S. Seara, M. Rust, and V. Vitelli, Information theory for data-driven model reduction in physics and biology, arXiv:2312.06608.
  25. Z. Ma and M. Collins, Noise contrastive estimation and negative sampling for conditional models: Consistency and statistical efficiency, in Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, edited by , , , (Association for Computational Linguistics, Brussels, Belgium, 2018), pp. 3698–3707.
  26. R. Kawai, J. M. R. Parrondo, and C. V. den Broeck, Dissipation: The phase-space perspective, Phys. Rev. Lett. 98, 080602 (2007).
  27. See Supplemental Material at http://link.aps.org/supplemental/10.1103/1ncx-6cxk for model and optimization details, and sufficiency proofs for the LENS formalism.
  28. Z. Rieder, J. L. Lebowitz, and E. Lieb, Properties of a harmonic crystal in a stationary nonequilibrium state, J. Math. Phys. 8, 1073 (1967).
  29. F. Bonetto, J. L. Lebowitz, and J. Lukkarinen, Fourier’s law for a harmonic crystal with self-consistent stochastic reservoirs, J. Stat. Phys. 116, 783 (2004).
  30. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization, Science 300, 2061 (2003).
  31. A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, Kinesin walks hand-over-hand, Science 303, 676 (2004).
  32. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, edited by H. Haken, Springer Series in Synergetics Vol.18 (Springer, Berlin, 1996).
  33. H. Chaté and P. Manneville, Phase diagram of the two-dimensional complex Ginzburg-Landau equation, Physica A 224, 348 (1996).
  34. I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys. 74, 99 (2002) .
  35. T. H. Tan, J. Liu, P. W. Miller, M. Tekant, J. Dunkel, and N. Fakhri, Topological turbulence in the membrane of a living cell, Nat. Phys. 16, 657 (2020).
  36. J. Li, C.-W. J. Liu, M. Szurek, and N. Fakhri, Measuring irreversibility from learned representations of biological patterns, PRX Life 2, 033013 (2024).
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/1ncx-6cxk for Movies M1 and M2.
  38. C. Joshi, S. Ray, L. M. Lemma, M. Varghese, G. Sharp, Z. Dogic, A. Baskaran, and M. F. Hagan, Data-driven discovery of active nematic hydrodynamics, Phys. Rev. Lett. 129, 258001 (2022).
  39. R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, Grad-CAM: Visual explanations from deep networks via gradient-based localization, in Proceedings of the 2017 IEEE International Conference on Computer Vision (ICCV) (IEEE, Piscataway, NJ, 2017), pp. 618–626.
  40. M. Watter, J. T. Springenberg, J. Boedecker, and M. Riedmiller, Embed to control: A locally linear latent dynamics model for control from raw images, in Proceedings of the 29th International Conference on Neural Information Processing Systems, edited by C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett (MIT Press, Cambridge, MA, 2015), Vol. 2, pp.2746–2754.
  41. A. v. d. Oord, Y. Li, and O. Vinyals, Representation learning with contrastive predictive coding, arXiv:1807.03748.
  42. Code repository for LENS, Catherine Ji, 2025, https://github.com/cj7280/LENS.

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Phys. Rev. Research 8, 023264– Published 8 June, 2026
  • Received 15 August 2025
  • Accepted 20 April 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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