[Paper Review] Thermodynamic speed limits from the regression of information

Schuyler B. Nicholson, Jason R. Green|arXiv (Cornell University)|May 4, 2021

Advanced Thermodynamics and Statistical Mechanics36 references4 citations

TL;DR

This paper derives thermodynamic speed limits and uncertainty relations from linear regression of information fluxes on nonequilibrium observables, showing that optimal linear predictors on statistical manifolds yield bounds on entropy production and system evolution speed. The key result is a multivariate time-information uncertainty relation that generalizes classical speed limits via Fisher information and covariance structures.

ABSTRACT

Irreversible processes accomplished in a fixed time involve nonlinearly coupled flows of matter, energy, and information. Here, using entropy production as an example, we show how thermodynamic uncertainty relations and speed limits on these nonlinear processes derive from linear regression. These uncertainty relations hold for both passive and actively-driven nonequilibrium processes and all have a mathematical form that mirrors uncertainty relations in quantum mechanics. Using optimal linear models, we show that information-theoretic variables naturally give physical predictions of the equation of motion on statistical manifolds in terms of physical observables. In these models, optimal intercepts are related to nonequilibrium analogs of Massieu functions/thermodynamic potentials, and optimal slopes are related to speed limits on collections of thermodynamic observables. Within this formalism, the second law of thermodynamics has a geometric interpretation as the nonnegativity of the slope and constrains the equation of motion. Overall, our results suggest that unknown relationships between nonequilibrium variables can be learned through statistical-mechanical inference.

Motivation & Objective

• To establish a formal link between linear regression of information fluxes and thermodynamic speed limits in nonequilibrium systems.
• To show that thermodynamic uncertainty relations emerge naturally from optimal linear prediction models on statistical manifolds.
• To generalize single-variable speed limits to multivariate cases using covariance between observables such as entropy and heat flux.
• To interpret the second law of thermodynamics geometrically as the nonnegativity of the regression slope in information content rate versus nonequilibrium deficiency.
• To enable inference of unknown relationships between nonequilibrium variables through statistical-mechanical regression.

Proposed method

• Uses linear regression to model the time evolution of information content rate $-\dot{I} = d\ln p/dt$ as a function of control variables $\bm{X}$, such as $I - \beta q$.
• Applies optimal linear prediction via minimizing mean squared error $\langle\bm{\mathcal{E}}^\top\bm{\mathcal{E}}\rangle$, yielding optimal intercept $\hat{a}$ and slope $\bm{\hat{b}}$.
• Derives the multivariate uncertainty relation from the determinant of the covariance matrix $|\bm{\Sigma}|$, which encodes correlations between observables.
• Expresses the speed limit using a cosine law form: $v_{I,q} = \sqrt{v_I^{-2} + v_q^{-2} - 2v_I^{-1}v_q^{-1}\cos\theta}$, where $\theta$ is the angle between fluctuations.
• Relates the optimal slope $\bm{\hat{b}}$ to the entropy production rate $\sigma$ via $\hat{b} = \sigma / \Delta X_{\omega}^2$, embedding thermodynamic constraints in regression coefficients.
• Uses Fisher information $I_F$ as the inverse of the variance of the optimal predictor, forming the basis of the time-information uncertainty relation $\tau^{-1} \geq |\sin\theta|^{-1} \tau_{I,q}^{-1}$.

Experimental results

Research questions

• RQ1 Can thermodynamic uncertainty relations and speed limits be derived from linear regression of information fluxes on nonequilibrium observables?
• RQ2 How do multivariate predictors—such as entropy and heat flux—improve bounds on system evolution speed compared to univariate models?
• RQ3 What is the geometric interpretation of the second law in terms of regression slope on statistical manifolds?
• RQ4 How do correlations between observables (e.g., $I$ and $q$) affect the tightness of thermodynamic speed limits?
• RQ5 Can statistical-mechanical inference through regression uncover unknown relationships between nonequilibrium variables?

Key findings

• The multivariate speed limit $\tau^{-1} \geq |\sin\theta|^{-1} \tau_{I,q}^{-1}$ is tighter than the univariate bound $\tau^{-1}_{\omega} = \sigma / \Delta(I - \beta q)$, especially when Fisher information changes rapidly.
• The optimal slope $\bm{\hat{b}} = \bm{\Sigma}^{-1} \langle \dot{S}/k_B, \beta\dot{Q} \rangle^\top$ is directly related to the entropy production rate and reflects the system's nonequilibrium response.
• The optimal intercept $\hat{a} = -\bm{\hat{b}}^\top \langle \mathbf{X} \rangle$ corresponds to the free entropy $\beta F$, linking regression to thermodynamic potentials.
• The second law emerges geometrically as the nonnegativity of the regression slope, with equilibrium and steady states having zero slope.
• When the Fisher information changes rapidly, the difference between univariate and bivariate speed bounds can exceed an order of magnitude, demonstrating the advantage of multivariate modeling.
• The time-information uncertainty relation is derived from fluctuations in the optimal linear predictor, showing that information geometry and thermodynamic bounds are fundamentally linked through regression.