In-vivo entropy production of A. subaru

Abstract: Entropy production is often used as a proxy for energy consumption of a non-equilibrium system. Lower bounds can be estimated from coarse-grained observations, and this has been done for various biological systems. Here, we apply these tools to a more macroscopic system whose true energy consumption is also known. We find that while entropy production does give a lower bound, it is some 25 orders of magnitude away from being saturated. To be certain of this result, we survey different methods of estimating irreversibility, and write down a novel kNN estimator.

• The paper demonstrates that coarse-grained entropy production in macroscopic systems, like A. subaru, severely underestimates energy dissipation by over 25 orders of magnitude.
• It employs various estimators—including Gaussian, discrete Markov models, kNN, and TUR frameworks—to quantify irreversibility using high-frequency OBD-II data.
• The work highlights the need for refined variable selection and methodological alignment to accurately capture functionally relevant energy consumption in complex systems.

In-vivo Entropy Production of A. subaru: An Analysis of Coarse-Grained Irreversibility in Macroscopic Systems

Introduction and Motivation

Entropy production is a foundational concept in stochastic thermodynamics, serving as a proxy for the energy dissipation required to sustain non-equilibrium states. It quantifies the irreversibility of physical processes, with tradition dictating that estimates can be made from partial or coarse-grained observational data. The primary theoretical relationship, $k_{B}T\leq W$, links the minimal energy dissipation required for an observed degree of irreversibility with the actual energy consumption $W$ of the underlying system. While this connection is routinely applied to microscopic and mesoscopic biological systems—where unresolved degrees of freedom limit our inference—the macroscopic application remains underexplored.

"In-vivo entropy production of A. subaru" (2604.00453) empirically interrogates the validity and implications of irreversibility measurements in a macroscopically driven, energetically well-characterized system: an automobile. By contrasting the entropy production rate accessible from macroscopic time series data with direct calorimetric estimates of energy consumption, the work critically assesses the informational content and practical relevance of irreversibility analyses when the microscopic underpinnings and functional aims of the observed system are manifest.

Experimental Framework and Estimation Techniques

The system under study, denoted "A. subaru," is instrumented via the automobile’s OBD-II interface, capturing two key variables—speed and engine RPM—at sub-second sampling frequency over extended operation. The coolant temperature, effectively constant over the snapshot, is discounted. These variables define a two-dimensional ($d=2$), coarse-grained state space.

A comprehensive suite of estimators—Gaussian process-based, discrete Markov models, kNN-driven estimators (including a newly introduced variant), and the thermodynamic uncertainty relation (TUR) framework—are evaluated:

• Gaussian Estimator: Assumes the process is a stationary Gaussian and utilizes spectral domain covariance structure, bias-corrected following Seara et al.
• Discrete Markov Chain: Employs empirical transitions between clustered states to estimate the net probability flux and thus irreversibility.
• kNN Estimators: Leverage nearest-neighbour counts/distances within both forward and time-reversed trajectory ensembles. Plug-in and KSG-style (count, union) estimators are articulated, adapting mutual information estimation strategies to the irreversibility setting.
• TUR-Based Bound: Implements the fluctuation-dissipation-informed thermodynamic uncertainty relation, choosing observables such as winding numbers to enforce an independent lower bound on entropy production.

Each methodology incorporates rigorous empirical estimation of bias (using data shuffling and synthetic noise), with hyper-parameter sensitivity (e.g., smoothing bandwidth, number of clusters, Markov lag order) systematically explored.

Principal Findings: Numerical Results and Bounds

The central result is unambiguous: the empirically accessible irreversibility from coarse-grained, macroscopic observables sets an exceedingly weak lower bound on energy dissipation—conservative by over 25 orders of magnitude. Specifically, the average entropy production rate is estimated at $\approx 0.5$ bits/s ($\sim 2 \times 10^{-21}$ J/s at room temperature), while the calorimetrically measured power consumption of the vehicle during steady-state operation is $W \approx 7 \times 10^4$ J/s. This enormous separation is robust across estimator choices and bias-correction protocols. The bound from the thermodynamic uncertainty relation is even weaker.

This gap far exceeds precedents from the biophysical literature, where neurons exhibit $\sim 8$ orders, auditory hair bundles $\sim 4$ orders, and microtubular dynamics similarly large but smaller gaps between the lower bound and actual energy dissipation.

Additional analyses indicate that moving to higher-order Markov models (i.e., incorporating longer history) leads to a nearly linear increase in estimated irreversibility, limited by data size and computational cost for discrete approaches, but kNN methods remain tractable.

Theoretical and Practical Implications

The analysis clarifies that for complex macroscopic systems, especially those with distinct functional aims (e.g., mechanical propulsion), irreversibility measured from typical observable time series becomes essentially decoupled from the system’s primary energetic expenditures. The operational degrees of freedom (engine, drivetrain) optimized for locomotive efficacy are not adequately sampled by the restricted, coarse snapshot available.

This raises the principle that, for broadly applicable biological and engineered systems, usefulness of irreversibility as a lower bound on energy consumption depends critically on whether the observed variables are mechanistically central to the function or merely incidental. In contexts (e.g., neuronal signaling, circadian clocks) where all relevant expenditure is funneled through the observed output, the bound is instructive or at minimum, interpretable. For systems whose energy consumption is dominated by unobserved, functionally distinct processes (e.g., propulsion through a resisting medium), coarse-grained irreversibility is at best a qualitative summary statistic, not a quantitative thermodynamic constraint.

The results thus underscore the necessity of aligning observational windows with the intrinsic information-processing or mechanical function of the system under interrogation when seeking to draw meaningful conclusions about energy-informational tradeoffs from irreversibility measurements.

Future Directions

Several research trajectories are suggested:

• Refined Variable Selection: Identifying or engineering observable sets that more faithfully capture the full set of non-equilibrium, function-driving degrees of freedom in macroscopic systems could potentially tighten these dramatic gaps.
• Entropy Production in Functionally Relevant Coordinates: For systems with complex hierarchical organization (e.g., vehicular traffic, flocking), development of statistical analysis aligned with mesoscale collective variables may yield more pertinent lower bounds.
• Benchmarking Statistical Estimators: The performance comparison of different estimators in large, structured, noisy datasets provides a practical reference for future methodological development and validation of irreversibility quantification.
• Application to Biological Systems: These gaps invite parallel investigation in multicellular and cognitive systems where mapping the relevant degrees of freedom onto observable proxies remains a central challenge.

Conclusion

This work comprehensively demonstrates that, for macroscopic systems with high-dimensional, functionally dissociated degrees of freedom, lower bounds on energy dissipation obtained from coarse-grained irreversibility are extraordinarily loose—by over 25 orders of magnitude in the studied case of A. subaru. As such, naive extrapolation of statistical thermodynamic bounds from observable time series is rarely informative in these contexts. The findings encourage a nuanced application of irreversibility-based analyses, emphasizing alignment with system functional objectives and careful interpretation of empirical lower bounds within the hierarchy of biological and physical organization.