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Universal tradeoff relation between speed, uncertainty, and dissipation in nonequilibrium stationary states

by Izaak Neri

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 12, 139 (2022)

Submission summary

As Contributors: Izaak Neri
Preprint link: scipost_202103_00027v2
Date submitted: 2021-09-28 19:08
Submitted by: Neri, Izaak
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Statistical and Soft Matter Physics
Approach: Theoretical

Abstract

We derive universal thermodynamic inequalities that bound from below the moments of first-passage times of stochastic currents in nonequilibrium stationary states and in the limit where the thresholds that define the first-passage problem are large. These inequalities describe a tradeoff between speed, uncertainty, and dissipation in nonequilibrium processes, which are quantified, respectively, with the moments of the first-passage times of stochastic currents, the splitting probability, and the mean entropy production rate. Near equilibrium, the inequalities imply that mean-first passage times are lower bounded by the Van't Hoff-Arrhenius law, whereas far from thermal equilibrium the bounds describe a universal speed limit for rate processes. When the current is the stochastic entropy production, then the bounds are equalities, a remarkable property that follows from the fact that the exponentiated negative entropy production is a martingale.

Current status:
Has been resubmitted


List of changes

The main changes are the following:

*) The Appendix B from the old manuscript, which was necessary to derive condition (26), has been removed.
Instead, the derivation in the new version of the manuscript relies on the condition (12), which states that J converges asymptotically to a drift-diffusion process, and the first-passage duality of a drift-diffusion process as derived in the new Appendix B.
The new derivation of the inequality is much simpler and more transparent [it relies on the condition (12) that is valid whenever J has finite memory].

*) Section 6 is novel. It provides an alternative derivation of the main results based on resulst from sequential hypothesis testing.

*) The introduction and discussion sections have been simplified, hopefully addressing more clearly the main points of the paper

*) The title has changed. Since the derived results express a tradeoff betweeen speed, uncertainty and dissipation, I thought that the present title better reflects the main point of the paper.

*) Several minor changes addressing the Referee's comments throughout the text.


Reports on this Submission

Anonymous Report 2 on 2021-12-10 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202103_00027v2, delivered 2021-12-10, doi: 10.21468/SciPost.Report.4021

Report

Referee report
I. Neri, "Universal tradeoff relation between speed, uncertainty, and dissipation in nonequilibrium stationary states"

This manuscript concerns the statistics of currents that can be observed in non-equilibrium systems in a steady state. This statistics is then put in context with the overall dissipation of the system. Such considerations are currently a hot topic, with the thermodynamic uncertainty relation (TUR) being the most prominent result. The present work goes beyond the standard TUR: instead of the variance of a current it characterizes the mean first passage time for the current to leave a set interval. The main result is formulated as an inequality, which becomes an equality when the current of interest is proportional to the entropy production. The resulting trade-off relation can, in some contexts, be interpreted as an extension of the Arrhenius law.

I think the manuscript is well written. The exposition of the main results in section 2 gives a useful overview, and the following derivation is thorough. The result opens a new pathway for experimental applications and further theoretical work, focusing on the interplay between first passage fluctuations and the energetics of nonequilibrium systems. Moreover, the tools used for the derivation, large deviation theory, martingale theory, and sequential hypothesis testing, complement each other and are of interest in their own right, providing a novel link between different research areas.

The manuscript has already successfully passed two rounds of refereeing, and I it should now be ready for acceptance.

One suggestion: In the introductory section (or the examples of Secs. 8 and 9), it might be helpful to discuss practical issues with the application of the relation. When taking the limit $\ell_-$ to infinity, the probability to leave the interval on the left boundary decreases exponentially. What would determine a good choice for a finite $\ell_-$, given a limited sampling capacity?

In addition, I have a few technical comments:

Below (19), $j_{ss}$ is introduced as "stationary probability flux", but it may not be obvious how this is defined.

For (29), it may be worth reminding the reader that the limit $\ell_-$ to infinity is taken, such that the saddle point approximation ("taking the maximum of the exponent") applies.

Typo "the the" before (40)

Below (40): "which clarifies the notation of Eq. (92)" it is somewhat strange to discuss this here, when the notation appears so much later. Maybe it would be better to refer back to this around (92). Or explain specifically what notation will be used in (92), without referring to the equation yet.

In the statement of Theorem 1, state which values the variable r can take (I assume integers or positive reals)

Last paragraph of p. 15: I think $z\approx 0$ should instead read $z\approx \bar j$ (1st instance) and $z\approx \dot s$ (2nd instance). Also, "or loose"->"are loose"

Fig. 5: In the legend, write "uncertainty relation" (or "TUR") instead of "uncertainty" (otherwise one could think the graph shows the relative uncertainty or similar).

  • validity: -
  • significance: -
  • originality: -
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Author:  Izaak Neri  on 2022-02-09  [id 2177]

(in reply to Report 2 on 2021-12-10)

I would like to thank the Referee for commenting positively on the manuscript and providing several useful comments.

Below you can find a point-by-point reply:

*) Referee: "One suggestion: In the introductory section (or the examples of Secs. 8 and 9), it might be helpful to discuss practical issues with the application of the relation. When taking the limit ℓ− to infinity, the probability to leave the interval on the left boundary decreases exponentially. What would determine a good choice for a finite ℓ−, given a limited sampling capacity?"

Reply:
This is indeed an important issue and should be addressed at some point. However, it concerns the practical issue of estimating the quantities in the inequality (3) and not the derivation of this inequality. When writing the paper I have made the decision to focus on theory. The paper derives the main results (3) and (6) and discusses their physical relevance. This is not to say that practical implementation of these inequalities in experiments are not important, but it would require an in-depth study on several examples. At present, the number of examples provided in the paper are too limited to address questions of the form "what is a good choice of $\ell_-$". Nevertheless, I will address this in a future work that focuses on practical questions, and for now, I have added the following sentence in the conclusion:

" In particular, the probability $p_-$ decreases exponentially with $\ell_-$, which raises the question how $p_-$ can be estimated at large values of $\ell_-$."


*) Referee: Below (19), jss is introduced as "stationary probability flux", but it may not be obvious how this is defined.

Reply: The equation (19) has been removed altogether as it is not used.

*) Referee: For (29), it may be worth reminding the reader that the limit ℓ− to infinity is taken, such that the saddle point approximation ("taking the maximum of the exponent") applies.

Reply: Absolutely. This reads now (22) and I have added:

“ For large values of $\ell_-$, the expression Eq.~(22) is a saddle point integral, and hence it is determined by the maximum of the exponent, i.e.,”

*) Referee: Typo "the the" before (40)

Reply: Fixed!

*) Referee: Below (40): "which clarifies the notation of Eq. (92)" it is somewhat strange to discuss this here, when the notation appears so much later. Maybe it would be better to refer back to this around (92). Or explain specifically what notation will be used in (92), without referring to the equation yet.

Answer: This was a typo (overlapping references). Eq.(92) should have read (40), and this has been fixed.

*) Referee: In the statement of Theorem 1, state which values the variable r can take (I assume integers or positive reals)

Reply: Good point. The values of r are integers, and this is now mentioned.

*) Referee: Last paragraph of p. 15: I think z≈0
should instead read z≈¯j (1st instance) and z≈˙s (2nd instance). Also, "or loose"->"are loose"

Reply: The referee is correct. I have fixed this.

*) Referee: Fig. 5: In the legend, write "uncertainty relation" (or "TUR") instead of "uncertainty" (otherwise one could think the graph shows the relative uncertainty or similar).

Reply: Fair point. Has been modified as suggested by the Referee.

Anonymous on 2022-02-10  [id 2182]

(in reply to Izaak Neri on 2022-02-09 [id 2177])

I have no objections to the answers given by the author and the changes made in the manuscript. I therefore agree that it should now be ready for acceptance.

Anonymous Report 1 on 2021-11-17 (Invited Report)

Report

I am satisfied with the response of the author and
the corresponding changes.

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  • significance: -
  • originality: -
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