Using precision coefficients on recurrence times and integrated currents to lower bound the average dissipation rate

Alberto Garilli; Diego Frezzato

arxiv: 2512.02647 · v2 · pith:XBMX2T2Pnew · submitted 2025-12-02 · ❄️ cond-mat.stat-mech

Using precision coefficients on recurrence times and integrated currents to lower bound the average dissipation rate

Alberto Garilli , Diego Frezzato This is my paper

Pith reviewed 2026-05-25 07:06 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords Markov jump processesentropy production ratethermodynamic uncertainty relationrecurrence timesintegrated currentsdissipation boundsprecision coefficients

0 comments

The pith

An inequality bounds the stationary entropy production rate using the precision of an integrated current together with forward and backward recurrence time statistics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a transition-based formalism for continuous-time Markov jump processes on irreducible networks. It expresses the long-time precision of a single integrated current in terms of the precisions of the forward and backward recurrence times plus an effective affinity that encodes the thermodynamic drive along the observed channel. From this relation the authors derive a general inequality that lower-bounds the average entropy production rate by current fluctuations while also incorporating recurrence-time statistics. The bound can be saturated under conditions less restrictive than those required by the thermodynamic uncertainty relation. A reader would care because the result supplies a potentially tighter route to estimating dissipation from observable fluctuation data in nanoscale chemical and biological systems.

Core claim

For continuous-time Markov jump processes on irreducible networks with time-independent rate constants, the long-time precision of a single integrated current over an observable channel can be expressed in terms of the precisions of the recurrence times of the forward and backward jumps and of an effective affinity that captures the thermodynamic driving on that channel. This leads to a general inequality that links the stationary entropy production rate with the fluctuations of an integrated current while also incorporating the statistics of the forward and backward recurrence times; the inequality can be saturated in less restrictive conditions than the TUR.

What carries the argument

Transition-based formalism that writes long-time current precision in terms of forward and backward recurrence-time precisions plus an effective affinity.

If this is right

The stationary entropy production rate is lower-bounded by a combination of integrated-current fluctuations and recurrence-time precisions.
The bound incorporates the statistics of both forward and backward recurrence times.
Saturation is possible under conditions less restrictive than those needed for the TUR.
The result supplies a route to estimating or optimizing dissipation rates in out-of-equilibrium nanoscale systems from measurable fluctuation data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be tested on small networks where recurrence times are directly observable, such as single-molecule enzyme turnover experiments.
Hybrid bounds that combine the new inequality with existing TUR forms might yield still tighter estimates of dissipation.
If recurrence-time statistics prove easier to measure than full current trajectories, the inequality could become a practical inference tool in experimental settings.

Load-bearing premise

The systems are continuous-time Markov jump processes on irreducible networks with time-independent rate constants.

What would settle it

An exact calculation or long simulation on any irreducible Markov network in which the derived inequality is violated while the standard TUR holds would falsify the central claim.

Figures

Figures reproduced from arXiv: 2512.02647 by Alberto Garilli, Diego Frezzato.

**Figure 1.** Figure 1: FIG. 1. (a) An example of Markov jump process where only bidirectional transitions along the channel connecting states [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Plot of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Comparison between lower bounds for the entropy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

read the original abstract

For continuous-time Markov jump processes on irreducible networks with time-independent rate constants, we employ a transition-based formalism to express the long-time precision of a single integrated current over an observable channel in terms of precisions of the recurrence times of the forward and backward jumps, and of an effective affinity that captures the thermodynamic driving on that channel. This leads to a general inequality that, similarly to the well-known Thermodynamic Uncertainty Relation (TUR), links the stationary entropy production rate with the fluctuations of an integrated current, but also incorporates the statistics of the forward and backward recurrence times. Such inequality can be saturated in less restrictive conditions than the TUR, and potentially offers new opportunities for the optimization and design of biological and chemical out-of-equilibrium systems at the nanoscale.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This derives a TUR variant for CTMC that folds recurrence-time precisions into the current-fluctuation bound and claims easier saturation.

read the letter

The main takeaway is a new lower bound on stationary entropy production for continuous-time Markov jump processes on irreducible networks. It starts from a transition-based expression for the long-time precision of an integrated current, writes that precision in terms of forward and backward recurrence-time precisions plus an effective affinity, and converts the result into an entropy-production inequality that can saturate under conditions less restrictive than the usual TUR. The setting is standard and clearly stated: time-independent rates on finite irreducible graphs. The construction itself looks like a direct, incremental extension of existing fluctuation relations rather than a wholesale reinvention. What the paper does cleanly is keep the thermodynamic driving explicit through the affinity term while bringing waiting-time statistics into the precision coefficient; that step is logically consistent with the transition formalism and could matter when recurrence times are the observable quantity. The abstract and stress-test show no internal contradictions or hidden circularity in the claim. The soft spots are modest but real. The provided summary contains no explicit equations, no saturation examples on even a two-state or three-state network, and no comparison of numerical tightness against TUR on the same process. Without those, it is impossible to judge whether the less-restrictive saturation is actually realized in any nontrivial case or whether the bound remains useful once the recurrence-time precisions are estimated from finite data. If the full manuscript supplies only the formal derivation without checks, the practical payoff stays speculative. This is for people already working on thermodynamic uncertainty relations and dissipation inference from single-channel fluctuations. A reader who needs a new bounding technique that explicitly uses waiting-time statistics might extract something usable, but only after verifying the algebra and seeing at least one concrete saturation case. It is worth sending to peer review; the scoped derivation is specific enough that referees can check the steps without needing broad new machinery.

Referee Report

1 major / 0 minor

Summary. The manuscript develops a transition-based formalism for continuous-time Markov jump processes on irreducible networks with time-independent rate constants. It expresses the long-time precision of a single integrated current over an observable channel in terms of the precisions of forward and backward recurrence times together with an effective affinity that captures the thermodynamic driving. This yields a general inequality relating the stationary entropy production rate to fluctuations of the integrated current while incorporating recurrence-time statistics; the bound is claimed to be saturable under less restrictive conditions than the thermodynamic uncertainty relation (TUR).

Significance. If the derivation holds and the saturation claim is substantiated, the result supplies a new lower bound on average dissipation that augments the TUR by explicit use of recurrence-time precision coefficients. This could enable tighter or more readily achievable bounds in the analysis and design of nanoscale out-of-equilibrium systems in biology and chemistry. The explicit scoping to irreducible CTMCs with constant rates is a strength that keeps the formalism well-defined.

major comments (1)

[Abstract] The abstract supplies no equations, explicit derivation steps, error analysis, or saturation examples, preventing verification that the final bound is independent of post-hoc choices in the definition of the effective affinity or the recurrence-time precisions. The central claim is a derivation whose load-bearing steps cannot be checked from the provided information.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for recognizing the potential significance of the result if the derivation holds. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] The abstract supplies no equations, explicit derivation steps, error analysis, or saturation examples, preventing verification that the final bound is independent of post-hoc choices in the definition of the effective affinity or the recurrence-time precisions. The central claim is a derivation whose load-bearing steps cannot be checked from the provided information.

Authors: The abstract is a concise high-level summary and therefore contains no equations or derivation steps; this is standard. The full transition-based formalism, the exact expression relating the long-time precision of the integrated current to the precisions of the forward and backward recurrence times together with the effective affinity, the resulting inequality for the stationary entropy production rate, and the saturation analysis are all derived rigorously in the main text. The effective affinity is defined directly from the thermodynamic driving force on the chosen observable channel, and the recurrence times are the canonical forward and backward waiting times for jumps across that channel; neither quantity involves post-hoc choices. Saturation examples under conditions weaker than those of the TUR are presented explicitly. As the work is a deterministic mathematical derivation for irreducible CTMCs with constant rates, no separate error analysis appears. We are prepared to insert one key equation into the abstract if the editor requests it. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation is scoped to continuous-time Markov jump processes on irreducible networks and uses an explicit transition-based formalism to relate long-time current precision to forward/backward recurrence-time precisions plus an effective affinity, from which an entropy-production inequality follows. No equation reduces by construction to a fitted parameter or renamed input, no load-bearing premise rests on self-citation, and the central bound is obtained mathematically rather than by statistical forcing or ansatz smuggling. The listed assumptions are necessary for the formalism and are stated openly; the result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard assumptions of continuous-time Markov jump processes on irreducible finite networks with constant rates; no free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)

domain assumption The network is irreducible and the rate constants are time-independent.
Stated in the abstract as the setting for the transition-based formalism.
domain assumption A transition-based formalism exists that expresses long-time current precision in terms of recurrence-time precisions and effective affinity.
Invoked to derive the inequality; no further justification supplied in abstract.

pith-pipeline@v0.9.0 · 5654 in / 1339 out tokens · 30097 ms · 2026-05-25T07:06:34.890045+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

T∞ = lim t→∞ t P[N_t] = -1/(ϵ J) + c_0 (⟨τ_{α→β}|α⟩ + ⟨τ_{β→α}|β⟩) + (P[τ_{α→β}] - P[τ_{β→α}])/J
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

σ_ss ≥ 2 J coth^{-1}(J T_∞ - ΔP_τ) with saturation on unicyclic networks

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

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P. Pietzonka and F. Coghi, Thermodynamic cost for pre- cision of general counting observables, Phys. Rev. E109, 064128 (2024)

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Therefore, throughout the paper, we indicate with the symbol\the removal of off-diagonal elements of a rate matrix associated to observable transitions

As explained in the main text,Sis obtained from the rate matrixRby setting the matrix elements (α, β) and (β, α) to zero, without modifying its diagonal elements. Therefore, throughout the paper, we indicate with the symbol\the removal of off-diagonal elements of a rate matrix associated to observable transitions. This should not be confused with stalling...

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J. R. Moffitt and C. Bustamante, Extracting signal from noise: kinetic mechanisms from a Michaelis–Menten-like expression for enzymatic fluctuations, The FEBS journal 281, 498 (2014)

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T. Mishima, D. Gupta, Y. Nakayama, W. C. Wareham, T. Ohyama, D. A. Sivak, and S. Toyabe, Efficiently driv- ing F 1 molecular motor in experiment by suppressing nonequilibrium variation, Physical Review Letters135, 148402 (2025)

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[36]

The integrand in Eq

Case 1:x 0 =s(ℓ) Let us consider the casex 0 =s(ℓ). The integrand in Eq. (A4) becomes, for eachn, ρ(τ, ℓ;t n, ¯ℓ;t n−1, ¯ℓ;· · ·;t 1, ¯ℓ|s(ℓ)) =k(ℓ) h e−(τ−t n)S i s(ℓ)s(ℓ) k(¯ℓ) n nY i=1 h e−(ti−ti−1)S i s(¯ℓ)s(ℓ) ! ,(A5) witht 0 = 0. The Laplace transform ofe −tS reads Z ∞ 0 dt e−tue−tS = (uI +S) −1 =S(u) −1,(A6) with I the indentity matrix and where we...

work page
[37]

Case 2:x 0 =t(ℓ) Similarly to the previous case, the Laplace transform of Eq. A3 forx 0 =t(ℓ) =s( ¯ℓ) reads L{ρ(t, ℓ|t(ℓ)}(u) =k(ℓ)[S(u) −1]s(ℓ)s(¯ℓ) +k(ℓ)k( ¯ℓ)[S(u)−1]s(ℓ)s(ℓ)[S(u)−1]s(¯ℓ)s(¯ℓ) ∞X n=0 [S(u)−1]s(¯ℓ)s(ℓ) n k(¯ℓ)n (A11) Taking (dL(−u)/du)u=0 and using similar arguments to those used to get Eq. (A10) we obtain the average occurrence times c...

work page

[1] [1]

A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett.114, 158101 (2015)

work page 2015

[2] [2]

T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state cur- rent fluctuations, Physical Review Letters116, 120601 (2016)

work page 2016

[3] [3]

Pietzonka, F

P. Pietzonka, F. Ritort, and U. Seifert, Finite-time gen- eralization of the thermodynamic uncertainty relation, Physical Review E96, 012101 (2017)

work page 2017

[4] [4]

J. M. Horowitz and T. R. Gingrich, Proof of the finite- time thermodynamic uncertainty relation for steady- state currents, Physical Review E96, 020103 (2017)

work page 2017

[5] [5]

Liu and J

K. Liu and J. Gu, Dynamical activity universally bounds precision of response in Markovian nonequilibrium sys- tems, Communications Physics8, 62 (2025)

work page 2025

[6] [6]

Song and C

Y. Song and C. Hyeon, Thermodynamic uncertainty rela- tion to assess biological processes, The Journal of Chem- ical Physics154(2021)

work page 2021

[7] [7]

J. D. Mallory, O. A. Igoshin, and A. B. Kolomeisky, Do we understand the mechanisms used by biological sys- tems to correct their errors?, The Journal of Physical Chemistry B124, 9289 (2020)

work page 2020

[8] [8]

Pietzonka, A

P. Pietzonka, A. C. Barato, and U. Seifert, Universal bound on the efficiency of molecular motors, Journal of Statistical Mechanics: Theory and Experiment2016, 124004 (2016)

work page 2016

[9] [9]

Hwang and C

W. Hwang and C. Hyeon, Energetic costs, precision, and transport efficiency of molecular motors, The journal of physical chemistry letters9, 513 (2018)

work page 2018

[10] [10]

Dechant, Multidimensional thermodynamic uncer- tainty relations, Journal of Physics A: Mathematical and Theoretical52, 035001 (2018)

A. Dechant, Multidimensional thermodynamic uncer- tainty relations, Journal of Physics A: Mathematical and Theoretical52, 035001 (2018)

work page 2018

[11] [11]

Falasco and M

G. Falasco and M. Esposito, Dissipation-time uncertainty relation, Physical Review Letters125, 120604 (2020). 6

work page 2020

[12] [12]

A. Pal, S. Reuveni, and S. Rahav, Thermodynamic uncer- tainty relation for first-passage times on Markov chains, Phys. Rev. Res.3, L032034 (2021)

work page 2021

[13] [13]

Oberreiter, U

L. Oberreiter, U. Seifert, and A. C. Barato, Universal minimal cost of coherent biochemical oscillations, Physi- cal Review E106, 014106 (2022)

work page 2022

[14] [14]

Marsland III, W

R. Marsland III, W. Cui, and J. M. Horowitz, The ther- modynamic uncertainty relation in biochemical oscilla- tions, Journal of the Royal Society Interface16, 20190098 (2019)

work page 2019

[15] [15]

Kamijima, K

T. Kamijima, K. Funo, and T. Sagawa, Finite-time ther- modynamic bounds and trade-off relations for informa- tion processing, Physical Review Research7, 013329 (2025)

work page 2025

[16] [16]

Van der Meer, B

J. Van der Meer, B. Ertel, and U. Seifert, Thermody- namic inference in partially accessible Markov networks: A unifying perspective from transition-based waiting time distributions, Physical Review X12, 031025 (2022)

work page 2022

[17] [17]

P. E. Harunari, A. Garilli, and M. Polettini, Beat of a current, Phys. Rev. E107, L042105 (2023)

work page 2023

[18] [18]

Garilli, P

A. Garilli, P. E. Harunari, and M. Polettini, Fluctuation relations for a few observable currents at their own beat, Journal of Physics A: Mathematical and Theoretical57, 455003 (2024)

work page 2024

[19] [19]

Pietzonka and F

P. Pietzonka and F. Coghi, Thermodynamic cost for pre- cision of general counting observables, Phys. Rev. E109, 064128 (2024)

work page 2024

[20] [20]

Garilli and D

A. Garilli and D. Frezzato, Interrelation between preci- sions on integrated currents and on recurrence times in Markov jump processes, Physical Review E112, 044141 (2025)

work page 2025

[21] [21]

P. E. Harunari, A. Dutta, M. Polettini, and E. Rold´ an, What to learn from a few visible transitions’ statistics?, Phys. Rev. X12, 041026 (2022)

work page 2022

[22] [22]

Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Re- views of Modern physics48, 571 (1976)

J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Re- views of Modern physics48, 571 (1976)

work page 1976

[23] [23]

Therefore, throughout the paper, we indicate with the symbol\the removal of off-diagonal elements of a rate matrix associated to observable transitions

As explained in the main text,Sis obtained from the rate matrixRby setting the matrix elements (α, β) and (β, α) to zero, without modifying its diagonal elements. Therefore, throughout the paper, we indicate with the symbol\the removal of off-diagonal elements of a rate matrix associated to observable transitions. This should not be confused with stalling...

work page

[24] [24]

M. J. Schnitzer and S. Block, Statistical kinetics of pro- cessive enzymes, inCold Spring Harbor Symposia on Quantitative Biology, Vol. 60 (Cold Spring Harbor Lab- oratory Press, 1995) pp. 793–802

work page 1995

[25] [25]

J. R. Moffitt and C. Bustamante, Extracting signal from noise: kinetic mechanisms from a Michaelis–Menten-like expression for enzymatic fluctuations, The FEBS journal 281, 498 (2014)

work page 2014

[26] [26]

J. R. Moffitt, Y. R. Chemla, and C. Bustamante, Meth- ods in statistical kinetics, inMethods in enzymology, Vol. 475 (Elsevier, 2010) pp. 221–257

work page 2010

[27] [27]

(16) is tighter than the TUR, since 2Jcoth −1(JT ∞)>2/T ∞, when|JT ∞|>1

For the very special case ∆P τ = 0, the bound Eq. (16) is tighter than the TUR, since 2Jcoth −1(JT ∞)>2/T ∞, when|JT ∞|>1

work page

[28] [28]

Polettini and M

M. Polettini and M. Esposito, Effective thermodynam- ics for a marginal observer, Physical review letters119, 240601 (2017)

work page 2017

[29] [29]

Polettini and M

M. Polettini and M. Esposito, Effective fluctuation and response theory, Journal of Statistical Physics176, 94 (2019)

work page 2019

[30] [30]

Raghu and I

A. Raghu and I. Neri, Effective affinity for generic cur- rents in Markov processes, Journal of Statistical Physics 192, 50 (2025)

work page 2025

[31] [31]

Watanabe-Nakayama, S

T. Watanabe-Nakayama, S. Toyabe, S. Kudo, S. Sugiyama, M. Yoshida, and E. Muneyuki, Effect of external torque on the ATP-driven rotation of F1-ATPase, Biochemical and biophysical research communications366, 951 (2008)

work page 2008

[32] [32]

Mishima, D

T. Mishima, D. Gupta, Y. Nakayama, W. C. Wareham, T. Ohyama, D. A. Sivak, and S. Toyabe, Efficiently driv- ing F 1 molecular motor in experiment by suppressing nonequilibrium variation, Physical Review Letters135, 148402 (2025)

work page 2025

[33] [33]

J. O. Wirth, L. Scheiderer, T. Engelhardt, J. Engel- hardt, J. Matthias, and S. W. Hell, MINFLUX dissects the unimpeded walking of kinesin-1, Science379, 1004 (2023)

work page 2023

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Deguchi, M

T. Deguchi, M. K. Iwanski, E.-M. Schentarra, C. Hei- debrecht, L. Schmidt, J. Heck, T. Weihs, S. Schnorren- berg, P. Hoess, S. Liu,et al., Direct observation of motor protein stepping in living cells using MINFLUX, Science 379, 1010 (2023)

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D. Frezzato, Stationary Markov jump processes in terms of average transition times: setup and some inequalities of kinetic and thermodynamic kind, Journal of Physics A: Mathematical and Theoretical53, 365003 (2020). Appendix A: Proof of Eqs.(10)and(11) Let↑:α→βbe the transition fromαtoβthrough the chosen channel among the ones connectingαandβ, and let↓:β→...

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[36] [36]

The integrand in Eq

Case 1:x 0 =s(ℓ) Let us consider the casex 0 =s(ℓ). The integrand in Eq. (A4) becomes, for eachn, ρ(τ, ℓ;t n, ¯ℓ;t n−1, ¯ℓ;· · ·;t 1, ¯ℓ|s(ℓ)) =k(ℓ) h e−(τ−t n)S i s(ℓ)s(ℓ) k(¯ℓ) n nY i=1 h e−(ti−ti−1)S i s(¯ℓ)s(ℓ) ! ,(A5) witht 0 = 0. The Laplace transform ofe −tS reads Z ∞ 0 dt e−tue−tS = (uI +S) −1 =S(u) −1,(A6) with I the indentity matrix and where we...

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[37] [37]

Case 2:x 0 =t(ℓ) Similarly to the previous case, the Laplace transform of Eq. A3 forx 0 =t(ℓ) =s( ¯ℓ) reads L{ρ(t, ℓ|t(ℓ)}(u) =k(ℓ)[S(u) −1]s(ℓ)s(¯ℓ) +k(ℓ)k( ¯ℓ)[S(u)−1]s(ℓ)s(ℓ)[S(u)−1]s(¯ℓ)s(¯ℓ) ∞X n=0 [S(u)−1]s(¯ℓ)s(ℓ) n k(¯ℓ)n (A11) Taking (dL(−u)/du)u=0 and using similar arguments to those used to get Eq. (A10) we obtain the average occurrence times c...

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