Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Universal energy-speed-accuracy trade-offs in driven nonequilibrium systems (2402.17931v2)

Published 27 Feb 2024 in cond-mat.stat-mech

Abstract: Physical systems driven away from equilibrium by an external controller dissipate heat to the environment; the excess entropy production in the thermal reservoir can be interpreted as a "cost" to transform the system in a finite time. The connection between measure theoretic optimal transport and dissipative nonequilibrium dynamics provides a language for quantifying this cost and has resulted in a collection of "thermodynamic speed limits", which argue that the minimum dissipation of a transformation between two probability distributions is directly proportional to the rate of driving. Thermodynamic speed limits rely on the assumption that the target probability distribution is perfectly realized, which is almost never the case in experiments or numerical simulations. Here, we address the ubiquitous situation in which the external controller is imperfect. As a consequence, we obtain a lower bound for the dissipated work in generic nonequilibrium control problems that 1) is asymptotically tight and 2) matches the thermodynamic speed limit in the case of optimal driving. We illustrate these bounds on analytically solvable examples and also develop a strategy for optimizing minimally dissipative protocols based on optimal transport flow matching, a generative machine learning technique. This latter approach ensures the scalability of both the theoretical and computational framework we put forth. Crucially, we demonstrate that we can compute the terms in our bound numerically using efficient algorithms from the computational optimal transport literature and that the protocols that we learn saturate the bound.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (42)
  1. Building normalizing flows with stochastic interpolants. In The eleventh international conference on learning representations, ICLR 2023, kigali, rwanda, may 1-5, 2023. OpenReview.net, 2023. tex.bibsource: dblp computer science bibliography, https://dblp.org tex.timestamp: Fri, 30 Jun 2023 14:55:53 +0200.
  2. Refined Second Law of Thermodynamics for Fast Random Processes. Journal of Statistical Physics, 147(3):487–505, May 2012.
  3. Optimal Protocols and Optimal Transport in Stochastic Thermodynamics. Physical Review Letters, 106(25):250601, June 2011.
  4. Thermodynamic Uncertainty Relation for Biomolecular Processes. Phys. Rev. Lett., 114(15):158101, April 2015.
  5. Quantum Statistical Learning via Quantum Wasserstein Natural Gradient. Journal of Statistical Physics, 182(1):7, January 2021.
  6. Realization of a micrometre-sized stochastic heat engine. Nature Physics, 8(2):143–146, February 2012. Number: 2 Publisher: Nature Publishing Group.
  7. Deep learning probability flows and entropy production rates in active matter, September 2023.
  8. Neural ordinary differential equations. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in neural information processing systems, volume 31. Curran Associates, Inc., 2018.
  9. Matrix Optimal Mass Transport: A Quantum Mechanical Approach. IEEE Transactions on Automatic Control, 63(8):2612–2619, August 2018. Conference Name: IEEE Transactions on Automatic Control.
  10. Optimal Transport for Gaussian Mixture Models. IEEE Access, 7:6269–6278, 2019.
  11. Unified, Geometric Framework for Nonequilibrium Protocol Optimization. Physical Review Letters, 130(10):107101, March 2023. Publisher: American Physical Society.
  12. Ensuring thermodynamic consistency with invertible coarse-graining. The Journal of Chemical Physics, 158(12):124126, March 2023. Publisher: American Institute of Physics.
  13. Gavin E. Crooks. Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Physical Review E, 60(3):2721–2726, September 1999.
  14. Gavin E Crooks. Measuring Thermodynamic Length. Phys. Rev. Lett., 99(10):100602, September 2007.
  15. Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K. Q. Weinberger, editors, Advances in neural information processing systems 26, pages 2292–2300. Curran Associates, Inc., 2013.
  16. Geometric decomposition of entropy production in out-of-equilibrium systems. Physical Review Research, 4(1):L012034, March 2022. tex.ids= dechant_geometric_2021 arXiv: 2109.12817.
  17. Optimal Control of Nonequilibrium Systems through Automatic Differentiation. Physical Review X, 13(4):041032, November 2023. Publisher: American Physical Society.
  18. Dissipation Bounds All Steady-State Current Fluctuations. Phys. Rev. Lett., 116(12):120601, March 2016.
  19. Inferring dissipation from current fluctuations. J. Phys. A: Math. Theor., 50(18):184004, April 2017.
  20. C. Jarzynski. Targeted free energy perturbation. Physical Review E, 65(4):046122, April 2002. arXiv: cond-mat/0109461.
  21. The Variational Formulation of the Fokker–Planck Equation. SIAM Journal on Mathematical Analysis, 29(1):1–17, January 1998.
  22. Physics-informed machine learning. Nature Reviews Physics, 3(6):422–440, June 2021.
  23. Jorge Kurchan. Fluctuation theorem for stochastic dynamics. J. Phys. A, 31(16):3719–3729, 1998.
  24. A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics. Journal of Statistical Physics, 95(1):333–365, April 1999.
  25. Flow Matching for Generative Modeling. September 2022.
  26. Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow. September 2022.
  27. Jan Maas. Gradient flows of the entropy for finite Markov chains. Journal of Functional Analysis, 261(8):2250–2292, October 2011.
  28. Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019.
  29. Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance. Physical Review Research, 3(4):043093, November 2021. Publisher: American Physical Society.
  30. Radford M Neal. Annealed importance sampling. Statistics and Computing, 11:125–139, 2001.
  31. Computational optimal transport: With applications to data science. Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019.
  32. Multisample Flow Matching: Straightening Flows with Minibatch Couplings. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett, editors, International Conference on Machine Learning, ICML 2023, 23-29 July 2023, Honolulu, Hawaii, USA, volume 202 of Proceedings of Machine Learning Research, pages 28100–28127. PMLR, 2023.
  33. Geometric approach to optimal nonequilibrium control: Minimizing dissipation in nanomagnetic spin systems. Physical Review E, 95(1):012148, January 2017.
  34. Stochastic thermodynamics of chemical reaction networks. J. Chem. Phys., 126(4):044101, 2007.
  35. U. Seifert. Stochastic thermodynamics: Principles and perspectives. European Physical Journal B, 64(3-4):423–431, 2008.
  36. Thermodynamic metrics and optimal paths. Phys. Rev. Lett., 108(19):190602, May 2012.
  37. Generalized thermodynamics of phase equilibria in scalar active matter. Physical Review E, 97(2):020602, February 2018. Publisher: American Physical Society.
  38. Daniel W. Stroock. Logarithmic Sobolev inequalities for gibbs states. In Eugene Fabes, Masatoshi Fukushima, Leonard Gross, Carlos Kenig, Michael Röckner, Daniel W. Stroock, Gianfausto Dell’Antonio, and Umberto Mosco, editors, Dirichlet Forms: Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna, Italy, June 8–19, 1992, Lecture Notes in Mathematics, pages 194–228. Springer, Berlin, Heidelberg, 1993.
  39. Improving and generalizing flow-based generative models with minibatch optimal transport, October 2023. arXiv:2302.00482 [cs].
  40. Escorted free energy simulations. J. Chem. Phys., page 12, 2011.
  41. Thermodynamic unification of optimal transport: Thermodynamic uncertainty relation, minimum dissipation, and thermodynamic speed limits. Phys. Rev. X, 13:011013, Feb 2023.
  42. Housekeeping and excess entropy production for general nonlinear dynamics. Physical Review Research, 5(1):013017, January 2023.
Citations (1)

Summary

  • The paper quantifies trade-offs among energy dissipation, speed, and accuracy by deriving asymptotically tight bounds for driven nonequilibrium systems.
  • It integrates optimal transport theory with computational techniques to develop scalable models for high-dimensional control.
  • Empirical validations with analytical models confirm that the new bounds align with classical thermodynamic limits under optimal driving conditions.

Overview of the Paper on Universal Energy-Speed-Accuracy Trade-offs in Driven Nonequilibrium Systems

This paper by Klinger and Rotskoff explores the inherent trade-offs between energy dissipation, speed, and accuracy within nonequilibrium thermodynamic systems, specifically when these systems are driven by an external controller. The primary motivation stems from the basic understanding that physical systems which are forced out of equilibrium tend to dissipate energy as heat into the environment. This paper makes significant strides in quantifying this dissipation, providing a lower bound for the minimum dissipated work in nonequilibrium control problems, a notable advance considering the common scenario of imperfect control in real-world applications.

Key Contributions

The paper outlines several central contributions:

  1. Extension of Thermodynamic Speed Limits: Traditional thermodynamic speed limits establish a relationship between dissipation and the rate of driving under the (often unrealistic) assumption of perfect realization of target probability distributions. This research extends these speed limits, factoring in imperfections in control protocols, and offers asymptotically tight bounds which hold under broader circumstances.
  2. Theoretical Framework and Computational Approach: A compelling element of this paper is its synthesis of optimal transport theory with machine learning, specifically using generative approaches akin to flow matching. This methodology not only creates scalable computational models but also enables near-perfect control in high-dimensional systems.
  3. Empirical Validation with Analytical Examples: The paper provides solid analytical groundwork, including results from solvable models that underscore the applicability and validity of the theoretical bounds proposed. Notably, it demonstrates that under these refined formulations, the derived bounds match the traditionally understood thermodynamic limits under conditions of optimal driving – an important validation.
  4. Scope of Applicability: The tightness of the proposed bounds in diverse conditions propounds a universal energy-speed-accuracy (ESA) trade-off, which extends to even highly complex and high-dimension systems. This is a particularly salient point for evaluating real-world systems where such ideal control is seldom achieved.

Methodological Advancements

The authors employ advanced concepts from optimal transport theory, connecting these with stochastic thermodynamics to refine the understanding of dissipation in nonequilibrium-driven processes. Through these means, they have presented an integrated view where understanding dissipative systems becomes more approachable computationally due to recent gains in machine learning.

Moreover, the paper's illustration of these concepts via high-dimensional control examples underlines the computational prowess the chosen methods bestow, offering compelling solutions to historically intractable problems. The simulations are tailored to reflect real experimental conditions more accurately, providing valuable insights into practical implementations of this theoretical framework in fields spanning synthetic biology to process optimization in chemical engineering.

Implications and Future Directions

The implications of this research stretch across both theoretical advancements and practical implementations. The framework provided by Klinger and Rotskoff's research is a powerful tool that aids better model development for assessing nonequilibrium processes which are common in both industrial and biological applications.

Looking towards the future, this work hints at continued intersections of machine learning algorithms with physics-driven models, possibly simplifying the complexity that comes with analyzing multi-dimensional datasets inherent to physical systems. Further exploration might probe the implications of these ESA trade-offs in quantum systems or consider the uncertainty they might introduce in measuring thermodynamic distances under stochastic influences.

In conclusion, the paper delivers substantial advancements in the understanding of driven nonequilibrium systems, positing that the newly identified bounds can serve as guiding tools for optimizing system control while accounting for real-world limitations in system design and execution. The crossover between disparate academic fields promises rich areas for future research, hinting towards a metaphoric convergence similar to the symbiotic relationship between energy, speed, and accuracy within these systems.

X Twitter Logo Streamline Icon: https://streamlinehq.com