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arxiv: 2606.23619 · v1 · pith:BDZFNEL5new · submitted 2026-06-22 · ❄️ cond-mat.stat-mech · physics.bio-ph· physics.chem-ph

Thermodynamic inference from noisy single-molecule time series

Todd R. Gingrich , Oleg A. Igoshin , Anatoly B. Kolomeisky , Dmitrii E. Makarov This is my paper

Pith reviewed 2026-06-26 06:18 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-phphysics.chem-ph
keywords entropy productionsingle-molecule trajectoriesdetection noisenon-Markovian dynamicsthermodynamic inferencespatial resolutiontemporal resolutionarrow of time
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The pith

Both strategies for estimating entropy production from noisy single-molecule trajectories yield lower bounds on the true value

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

The paper investigates how finite spatial and temporal resolution in single-molecule measurements degrade one's ability to quantify entropy production, which measures the arrow of time. It compares two common strategies: first recovering the statistics of the underlying clean process X(t) from the noisy observable Y(t) before estimating entropy production, and second applying the estimator directly to Y(t) as a proxy. Both strategies are shown to produce values that are always less than or equal to the true entropy production. A reader would care because real experiments are always limited by diffraction and photon counting rates, so understanding these systematic underestimates affects how dissipation is inferred in molecular systems. The analysis further reveals that Y(t) follows non-Markovian dynamics even when X(t) is Markovian, and that spatial and temporal resolutions interact in a nontrivial way.

Core claim

Given an experimental time series Y(t) degraded by noise and corresponding to a microscopic variable X(t), both the strategy of inferring the statistical properties of X(t) from Y(t) before estimating entropy production and the strategy of applying the entropy production estimator directly to Y(t) result in lower bounds on the true entropy production. Noise-degraded observables Y(t) undergo non-Markovian dynamics even when X(t) are Markovian, and non-Markovian entropy production estimators are advantageous. There is a nontrivial interplay between spatial and temporal resolution: in the presence of detection noise, improving the temporal resolution alone may lead to poorer rather than better

What carries the argument

The two strategies for thermodynamic inference from noisy observables Y(t), together with the proofs that each produces a lower bound on entropy production of the underlying X(t); the demonstration that Y(t) is non-Markovian whenever X(t) is Markovian.

If this is right

  • Non-Markovian entropy production estimators are advantageous over Markovian ones when applied to noise-degraded data.
  • Improving temporal resolution in isolation can degrade the quality of entropy production estimates when spatial detection noise is present.
  • Entropy production remains detectable from below even with severely limited experimental resolution.
  • Measurements obtained by either strategy can be interpreted as conservative lower bounds on actual dissipation.

Where Pith is reading between the lines

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

  • Experimental design may require joint optimization of spatial and temporal sampling rather than maximizing frame rate alone.
  • The lower-bound property could extend to inference of other nonequilibrium signatures from noisy single-particle tracks.
  • Targeted denoising algorithms might be developed to tighten these bounds on thermodynamic quantities specifically.

Load-bearing premise

The underlying microscopic dynamics X(t) are Markovian.

What would settle it

Simulate exact Markovian trajectories X(t) to compute their true entropy production, add realistic spatial and temporal noise to generate Y(t), apply both estimation strategies to the noisy series, and check whether the resulting values are always less than or equal to the true entropy production.

Figures

Figures reproduced from arXiv: 2606.23619 by Anatoly B. Kolomeisky, Dmitrii E. Makarov, Oleg A. Igoshin, Todd R. Gingrich.

Figure 1
Figure 1. Figure 1: For each we follow the same procedure 1. Generate noiseless trajectory with 2 × 107 transi￾tions using the Gillespie algorithm 2. Sample the trajectory at time intervals τ 3. Introduce measurement noise every time step 4. Compute transition probabilities from the uncor￾rected trajectories and estimate entropy production with the first-order Markov estimator 5. Compute the corrected entropy production using… view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Entropy production estimates calculations. (A-C) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: It should be noted that estimating Eq. (19) at [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Non-Markov entropy production estimates, Eq. (19), [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Entropy production estimate, Eq. (19), as a function [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Entropy production estimated from the photon color [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

Single-molecule or single-particle tracking measurements inherently yield noisy microscopic trajectories, often significantly constrained by the diffraction limit and by the finite rate at which photons are emitted and counted. Here we study systematically the resulting effects of finite spatial and temporal resolution on one's ability to discern and quantify the arrow of time in microscopic trajectories. Given an experimental time series Y(t) degraded by noise, we consider the problem of estimating the entropy production associated with the corresponding microscopic variable X(t) using two strategies. The first attempts to infer the statistical properties of X(t) from those of Y(t) before estimating the entropy production. The second uses the experimental observable as a proxy for the true microscopic observable, with the entropy production estimator applied directly to Y(t). We prove that both strategies result in lower bounds on the true entropy production. Importantly, noise-degraded observables Y(t) undergo non-Markovian dynamics even when X(t) are Markovian, and non-Markovian entropy production estimators are advantageous. We further note nontrivial interplay between spatial and temporal resolution: in the presence of detection noise, improving the temporal resolution alone may lead to poorer rather than better entropy production estimates.

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.

Referee Report

0 major / 1 minor

Summary. The manuscript examines how finite spatial and temporal resolution in single-molecule tracking introduce noise into observed trajectories Y(t) derived from underlying Markovian microscopic dynamics X(t). It analyzes two inference strategies for entropy production: (i) first recovering statistical properties of X(t) from Y(t) then estimating entropy production, and (ii) applying entropy production estimators directly to the noisy Y(t). The central claims are that both strategies produce rigorous lower bounds on the true entropy production, that Y(t) is non-Markovian even when X(t) is Markovian (making non-Markovian estimators advantageous), and that there is a nontrivial interplay between spatial and temporal resolution such that improving temporal resolution alone can worsen entropy production estimates under detection noise.

Significance. If the stated proofs hold, the work supplies parameter-free lower bounds on entropy production together with falsifiable predictions about estimator performance and resolution trade-offs. These results are significant for stochastic thermodynamics and single-molecule experiments, as they directly address practical limitations in quantifying the arrow of time from noisy data and credit the non-Markovian character of the observable.

minor comments (1)
  1. The abstract and introduction state the Markovian assumption for X(t) explicitly, but a dedicated section clarifying the precise noise model (e.g., additive Gaussian detection noise versus photon-counting statistics) would aid reproducibility of the derivations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of our central claims and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; central claims are explicit mathematical proofs from stated assumptions

full rationale

The manuscript derives lower bounds on entropy production via two inference strategies applied to noisy observables Y(t) when the underlying X(t) is Markovian. These are presented as direct proofs under the given noise model, without parameter fitting, self-referential definitions, or load-bearing self-citations. The non-Markovian character of Y(t) and the resolution-interplay result follow from the same first-principles stochastic-process arguments. The derivation chain is self-contained against external benchmarks and does not reduce any prediction to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on standard stochastic thermodynamics definitions plus a domain assumption about underlying Markovian dynamics and a specific experimental noise model; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The true microscopic dynamics X(t) are Markovian.
    Invoked when stating that Y(t) undergoes non-Markovian dynamics even when X(t) are Markovian.

pith-pipeline@v0.9.1-grok · 5753 in / 1089 out tokens · 33699 ms · 2026-06-26T06:18:29.809578+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

55 extracted references

  1. [1]

    Thus a first-order Markovian estimate suffices

    IfP(Y|X) =δ Y X (i.e., no errors in determining the stateX) then the sequence of observed statesX k is Markovian, and⟨ ˙S⟩k(τ) is the same for anyk≥1. Thus a first-order Markovian estimate suffices

  2. [2]

    In the limitτ→0 andP(Y|X)→δ Y X this esti- mator will give the exact entropy production rate for anyk≥1,⟨ ˙S⟩k(τ)→ ⟨ ˙S⟩

  3. [3]

    PROOF: Since the discrete-time dynamics ofXis Markovian, the estimated entropy production can be written as ⟨ ˙S⟩k(τ) =⟨ ˙S⟩1(τ) = 1 τ X X0,X1 P(X 1, X0) log P(X 1, X0) P(X 0, X1)

    In the caseP(Y|X) =δ Y X we have the inequality ⟨ ˙S⟩k ≤ ⟨ ˙S⟩, which becomes identity whenτ→0. PROOF: Since the discrete-time dynamics ofXis Markovian, the estimated entropy production can be written as ⟨ ˙S⟩k(τ) =⟨ ˙S⟩1(τ) = 1 τ X X0,X1 P(X 1, X0) log P(X 1, X0) P(X 0, X1) . (20) LetX 1(0),0(1)(t) denote any microscopic path sat- isfying the conditionsX...

  4. [4]

    In other words, experi- mental errors in measuringX(t) or finite time res- olution cannot introduce spurious irreversibility in Y(t)

    If the microscopic trajectoryX(t) satisfies detailed balance then⟨ ˙S⟩k(τ) = 0. In other words, experi- mental errors in measuringX(t) or finite time res- olution cannot introduce spurious irreversibility in Y(t). PROOF: Using Eq. (10), one finds that, if the detailed balance conditionP(X ′, τ;X,0) = P(X, τ;X ′,0) is satisfied forX(t), it is also sat- isf...

  5. [5]

    Markov estimate of the entropy production from Y(t) is always a lower bound on that estimated fromX(t). More precisely, 1 τ X X0,X1 P(X 1, X0) log P(X 1, X0) P(X 0, X1) ≥ 1 τ X Y0,Y1 P(Y 1, Y0) log P(Y 1, Y0) P(Y 0, Y1) (21) PROOF: First, let us prove the following auxiliary statement, which is a form of the data processing inequality. Letxbe a random vec...

  6. [6]

    The entropy production estimated from sequencesY 0, Y1

    The above argument can be extended to higher- order entropy production estimates. The entropy production estimated from sequencesY 0, Y1 . . . Yk will always be lower than or equal to that from sequencesX 0, X1 . . . Xk. But the latter does not depend onksinceXis Markovian. We conclude that the entropy production estimated fromY i to arbitrary Markov orde...

  7. [7]

    Generate noiseless trajectory with 2×10 7 transi- tions using the Gillespie algorithm

  8. [8]

    Sample the trajectory at time intervalsτ

  9. [9]

    Introduce measurement noise every time step

  10. [10]

    Compute transition probabilities from the uncor- rected trajectories and estimate entropy production with the first-order Markov estimator

  11. [11]

    microscopic rate matrix

    Compute the corrected entropy production using the recovered transition-rate matrix computed with Eq. (18). First, we consider a walker on a ring, i.e., a simple 1-D chemical reaction network with periodic boundary conditions (Fig. 1A), 1↔2↔ · · · ↔N↔1. This network can describe, for instance, the sequence of rota- tional states of a rotary molecular moto...

  12. [12]

    Peliti and S

    L. Peliti and S. Pigolotti,Stochastic Thermodynamics: An Introduction(Princeton University Press, 2021)

    2021

  13. [13]

    Seifert,Stochastic Thermodynamics(Cambridge Uni- versity Press, Cambridge, 2025)

    U. Seifert,Stochastic Thermodynamics(Cambridge Uni- versity Press, Cambridge, 2025)

    2025

  14. [14]

    Rold´ an and J

    E. Rold´ an and J. M. R. Parrondo, Entropy production and Kullback-Leibler divergence between stationary tra- jectories of discrete systems, Phys. Rev. E85, 031129 (2012)

    2012

  15. [15]

    Hartich and A

    D. Hartich and A. Godec, Violation of local detailed bal- ance upon lumping despite a clear timescale separation, Phys. Rev. Res.5, L032017 (2023)

    2023

  16. [16]

    van der Meer, J

    J. van der Meer, J. Deg¨ unther, and U. Seifert, Time- resolved statistics of snippets as general framework for model-free entropy estimators, Phys. Rev. Lett.130, 257101 (2023)

    2023

  17. [17]

    van der Meer, B

    J. van der Meer, B. Ertel, and U. Seifert, Thermodynamic inference in partially accessible Markov networks: A uni- fying perspective from transition-based waiting time dis- tributions, Phys. Rev. X12, 031025 (2022)

    2022

  18. [18]

    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)

    2022

  19. [19]

    J. Li, J. M. Horowitz, T. R. Gingrich, and N. Fakhri, Quantifying dissipation using fluctuating currents, Nat. Commun.10, 1666 (2019)

    2019

  20. [20]

    J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctua- tions, Nat. Phys.16, 15 (2019)

    2019

  21. [21]

    Bisker, M

    G. Bisker, M. Polettini, T. R. Gingrich, and J. M. Horowitz, Hierarchical bounds on entropy production in- ferred from partial information, J. Stat. Mech.2017, 093210 (2017)

    2017

  22. [22]

    T. R. Gingrich, G. M. Rotskoff, and J. M. Horowitz, In- ferring dissipation from current fluctuations, J. Phys. A: Math. Theor.50, 184004 (2017)

    2017

  23. [23]

    T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluctuations, Phys. Rev. Lett.116, 120601 (2016)

    2016

  24. [24]

    P. E. Harunari, Uncovering nonequilibrium from unre- solved events, Phys. Rev. E110, 024122 (2024)

    2024

  25. [25]

    Godec and D

    A. Godec and D. E. Makarov, Challenges in inferring the directionality of active molecular processes from single- molecule fluorescence resonance energy transfer trajecto- ries, J. Phys. Chem. Lett.14, 49 (2023). 14

    2023

  26. [26]

    J. H. Fritz, B. Ertel, and U. Seifert, Entropy estimation for partially accessible Markov networks based on im- perfect observations: Role of finite resolution and finite statistics, Phys. Rev. E111, 044106 (2025)

    2025

  27. [27]

    M. T. Bauer, U. Seifert, and J. van der Meer, Strobo- scopic measurements in Markov networks: exact genera- tor reconstruction vs. thermodynamic inference, J. Phys. A: Math. Theor.58, 125001 (2025)

    2025

  28. [28]

    K. Song, D. E. Makarov, and E. Vouga, Information- theoretical limit on the estimates of dissipation by molec- ular machines using single-molecule fluorescence reso- nance energy transfer experiments, J. Chem. Phys.161, 044111 (2024)

    2024

  29. [29]

    Yu and P

    Q. Yu and P. E. Harunari, Dissipation at limited reso- lutions: power law and detection of hidden dissipative scales, Journal of Statistical Mechanics: Theory and Ex- periment2024, 103201 (2024)

    2024

  30. [30]

    Fazel, K

    M. Fazel, K. S. Grussmayer, B. Ferdman, A. Radenovic, Y. Shechtman, J. Enderlein, and S. Press´ e, Fluorescence microscopy: A statistics-optics perspective, Rev. Mod. Phys.96, 025003 (2024)

    2024

  31. [31]

    Hartich and A

    D. Hartich and A. Godec, Emergent memory and kinetic hysteresis in strongly driven networks, Phys. Rev. X11, 041047 (2021)

    2021

  32. [32]

    Esposito, Stochastic thermodynamics under coarse graining, Phys

    M. Esposito, Stochastic thermodynamics under coarse graining, Phys. Rev. E85, 041125 (2012)

    2012

  33. [33]

    Rahav and C

    S. Rahav and C. Jarzynski, Fluctuation relations and coarse-graining, J. Stat. Mech.2007, P09012 (2007)

    2007

  34. [34]

    K. Blom, K. Song, E. Vouga, A. Godec, and D. E. Makarov, Milestoning estimators of dissipation in sys- tems observed at a coarse resolution, Proc. Natl. Acad. Sci. U.S.A.121, e2318333121 (2024)

    2024

  35. [35]

    G´ omez-Mar´ ın, J

    A. G´ omez-Mar´ ın, J. M. R. Parrondo, and C. Van den Broeck, Lower bounds on dissipation upon coarse grain- ing, Phys. Rev. E78, 011107 (2008)

    2008

  36. [36]

    This case can be considered using the same formalism, as will be seen below

    In fluorescence experiments, photons arrive at random times, with only the average timeτknown. This case can be considered using the same formalism, as will be seen below

  37. [37]

    Zwanzig,Nonequilibrium Statistical Mechanics(Ox- ford University Press, 2001)

    R. Zwanzig,Nonequilibrium Statistical Mechanics(Ox- ford University Press, 2001)

    2001

  38. [38]

    Note that this is a deconvolution problem, which is often notoriously difficult numerically

  39. [39]

    Oster and H

    G. Oster and H. Wang, Rotary protein motors, Trends Cell Biol.13, 114 (2003)

    2003

  40. [40]

    A. B. Kolomeisky,Motor Proteins and Molecular Motors (CRC Press, 2015) p. 222

    2015

  41. [41]

    O. A. Igoshin, A. B. Kolomeisky, and D. E. Makarov, Uncovering dissipation from coarse observables: A case study of a random walk with unobserved internal states, J. Chem. Phys.162, 034111 (2025)

    2025

  42. [42]

    Rold´ an and J

    E. Rold´ an and J. M. R. Parrondo, Estimating dissipation from single stationary trajectories, Phys. Rev. Lett.105, 150607 (2010)

    2010

  43. [43]

    Haran and H

    G. Haran and H. Hofmann, Single-molecule fluores- cence spectroscopy of fast protein dynamics, Curr. Opin. Struct. Biol.98, 103242 (2026)

    2026

  44. [44]

    Vollmar, J

    L. Vollmar, J. Schimpf, B. Hermann, and T. Hugel, Cochaperones convey the energy of ATP hydrolysis for directional action of Hsp90, Nat. Commun.15, 569 (2024)

    2024

  45. [45]

    Ratzke, B

    C. Ratzke, B. Hellenkamp, and T. Hugel, Four-colour FRET reveals directionality in the Hsp90 multicompo- nent machinery, Nat. Commun.5, 4192 (2014)

    2014

  46. [46]

    A. B. Kolomeisky, Motor proteins and molecular motors: how to operate machines at the nanoscale, J. Phys.: Con- dens. Matter25, 463101 (2013)

    2013

  47. [47]

    Koza, General technique of calculating the drift veloc- ity and diffusion coefficient in arbitrary periodic systems, J

    Z. Koza, General technique of calculating the drift veloc- ity and diffusion coefficient in arbitrary periodic systems, J. Phys. A: Math. Gen.32, 7637 (1999)

    1999

  48. [48]

    Koza, Diffusion coefficient and drift velocity in peri- odic media, Phys

    Z. Koza, Diffusion coefficient and drift velocity in peri- odic media, Phys. Rev. E65, 031905 (2002)

    2002

  49. [49]

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

    2015

  50. [50]

    I. V. Gopich and A. Szabo, Decoding the pattern of pho- ton colors in single-molecule FRET, J. Phys. Chem. B 113, 10965 (2009)

    2009

  51. [51]

    I. V. Gopich, Accuracy of maximum likelihood estimates of a two-state model in single-molecule FRET, J. Chem. Phys.142, 034110 (2015)

    2015

  52. [52]

    Press´ e and I

    S. Press´ e and I. Sgouralis,Data Modeling for the Sci- ences: Applications, Basics, Computations(Cambridge University Press, 2023)

    2023

  53. [53]

    Saurabh, M

    A. Saurabh, M. Fazel, M. Safar, I. Sgouralis, and S. Press´ e, Single-photon smFRET. I: Theory and con- ceptual basis, Biophys. Rep.3, 100089 (2023)

    2023

  54. [54]

    Saurabh, M

    A. Saurabh, M. Safar, M. Fazel, I. Sgouralis, and S. Press´ e, Single-photon smFRET: II. application to con- tinuous illumination, Biophys. Rep.3, 100087 (2023)

    2023

  55. [55]

    Schweiger, A

    M. Schweiger, A. Saurabh, and S. Press´ e, A cautious user’s guide in applying HMMs to physical systems, J. Chem. Phys.163, 210901 (2025)

    2025