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

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.

• 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.