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Extreme value statistics of edge currents in Markov jump processes and their use for entropy production estimation
by Izaak Neri, Matteo Polettini
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Submission summary
Authors (as registered SciPost users):  Izaak Neri · Matteo Polettini 
Submission information  

Preprint Link:  https://arxiv.org/abs/2208.02839v3 (pdf) 
Date accepted:  20230330 
Date submitted:  20230112 06:30 
Submitted by:  Neri, Izaak 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The infimum of an integrated current is its extreme value against the direction of its average flow. Using martingale theory, we show that the infima of integrated edge currents in timehomogeneous Markov jump processes are geometrically distributed, with a mean value determined by the effective affinity measured by a marginal observer that only sees the integrated edge current. In addition, we show that a marginal observer can estimate a finite fraction of the average entropy production rate in the underlying nonequilibrium process from the extreme value statistics in the integrated edge current. The estimated average rate of dissipation obtained in this way equals the above mentioned effective affinity times the average edge current. Moreover, we show that estimates of dissipation based on extreme value statistics can be significantly more accurate than those based on thermodynamic uncertainty ratios, as well as those based on a naive estimator obtained by neglecting nonMarkovian correlations in the KullbackLeibler divergence of the trajectories of the integrated edge current.
Author comments upon resubmission
Both Referees commented positively on the manuscript, for example, both Referees mention that the validity, significance and originality of the paper is high.
We have taken the suggestions of the Referees to improve the manuscript onboard, and we have replied to all their queries (see individual replies to the Referee's comments).
Having taken onboard all Referees' comments, we hope the present manuscript will be considered suitable for publication in SciPost Physics.
Kind regards,
Izaak Neri and Matteo Polettini
List of changes
To address the Referees' comments, we have made, amongst others, the following changes in the manuscript:
*) Following the First Referee's suggestion, we have added a paragraph in the introduction discussing the relation between the submitted manuscript and the paper [G. Bisker, M. Polettini, T. R. Gingrich and J. M. Horowitz, Hierarchical bounds on
entropy production inferred from partial information, Journal of Statistical Mechanics:
Theory and Experiment 2017(9), 093210 (2017)], which is Reference [35] in the submitted manuscript.
*) We have clarified in the abstract that estimates of dissipation based on infimum statistics are better than a naive estimator of the KullbackLeibler divergence that neglects nonMarkovian correlations in the trajectories of the current.
*) We have added the Appendix C that compares microscopic affinities with effective in unicyclic systems. This is to address a comment of Referee 2 regarding the relation between the microscopic affinity, the effective affinity, and the "true" macroscopic affinity.
*) We have added Appendix H that comments on the infimum statistics of currents that are not edge currents (as requested by the second Referee).
Published as SciPost Phys. 14, 131 (2023)
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2023216 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2208.02839v3, delivered 20230216, doi: 10.21468/SciPost.Report.6752
Report
I'm sorry for the delay, I had overlooked that this manuscript needed my input.
With the changes in the manuscript and the replies to the previous comments (almost) all questions have been clarified, and the manuscript should now be ready for publication.
Just a few optional points to consider, and responses to the answers to my previous points:
(d): I had kind of expected that proportionality is understood up to boundary terms. I'm not aware of any good examples of nonequilibrium systems where the entropy production is indeed proportional to a single edge current. This would mean that the entropy production of all other edges is zero exactly (otherwise the entropy would sometimes change without a change in Jxy). But if there is a current through one edge, there must also be a current though some other edges (by the Kirchhoff rules, see also point (f)), which comes with entropy production. The only example I can think of is a network with a single state and a single edge, leading from the state into itself. This somewhat stretches the present notation, where I believe x!=y is usually assumed out of notational convenience. If this asymmetric random walk is indeed the only relevant example, then I think Sec. 3 would profit from being specified for this system. On the other hand, the class of examples could be enlarged by adding deadend edges that carry no current. At least I'd like to see discussed that the class of systems where this proportionality is exact is quite limited.
(f): I appreciate the clarifications concerning the "driving by a single edge". However, I feel that Sec. 6.4 now carries very little information, if any. Isn't this just introducing more notation, essentially defining the "thermodynamic force" simply as the microscopic affinity divided by temperature? Either the fundamental relevance of this class of systems should be explained in Sec. 6.4, or the Section could be removed altogether, in favour of simply pointing out the possibility of driving by a single edge in the context of the examples.
(h): I was oblivious of the fact that the quadratic bound [Eq. (16) of the reply] could yield stronger bounds on the entropy production than the TUR, when evaluated at nontypical z. Thank you for pointing this out. So am I right to assume that Eq. (16), evaluated at the optimal z, would, in Fig. 5, perform just as well as (or even better than) the estimator based on the infimum statistics? (I don't expect the authors to add this to the plot.) Nonetheless, I see the advantages of their new formalism.
16. Thank you for this detailed and illuminating explanation. Though I feel I should have consulted a textbook rather than bothering the authors ...
Report #1 by Anonymous (Referee 3) on 2023126 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2208.02839v3, delivered 20230126, doi: 10.21468/SciPost.Report.6611
Strengths
see report
Weaknesses
see report
Report
The adjustments in this revised manuscript clarify most of my previously
raised concerns. Taking additionally into account the changes made based
on the report of referee 2, the present version of the manuscript
shows substantial improvements over the previous one. This work is now,
in my opinion, a valuable addition to the field highlighting the
operational significance of martingale theory for inference problems in
partially accessible Markov jump processes.
As a last remark, I have to iterate that the proposed entropy
estimator, dubbed the modified infimum ratio, is in fact the
"informed partial" estimator from Gili Bisker et al J. Stat. Mech.
(2017) 093210 and therefore not a novel entropy estimator. Thus, as the
authors have already acknowledged in their reply, the main result of
their work is neither the average or trajectorydependent entropy
estimator per se, nor a thermodynamic interpretation for the effective
affinity a of a transition in terms of "effective thermodynamics".
Rather, it is the successful application of martingales and the obtained
law for the infimum current which comprise the novel insights of this
work.
Arguably, this focus is not sufficiently emphasized in the present
version of the manuscript (especially in the abstract and introduction).
It becomes clear only after carefully comparing this work to the relevant
references.
Despite this minor shortcoming, I recommend publication of this work
in SciPost as now is.