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Perspective: Time irreversibility in systems observed at coarse resolution

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Pith's one-line read Milestoning and time reversal do not commute, so coarse-grained observations can show apparent entropy production even when the microscopic dynamics obeys detailed balance.

desk verdict A clear synthesis of why milestoning breaks time-reversal inference, but the internal contradiction about saturated-order estimators weakens its main remedy. read the letter →

arxiv 2412.02675 v3 pith:NDYXGXHZ submitted 2024-12-03 cond-mat.stat-mech cond-mat.softphysics.chem-ph

Cai Dieball , Aljaž Godec This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.softphysics.chem-ph MSC 82C3160J6060K15 PACS 05.40.-a05.70.Ln
keywords time-reversalsymmetrydetailedbalanceentropyproductioncoarsegrainingmilestoningkinetichysteresissemi-Markovprocessesstochasticthermodynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This perspective argues that the time-reversal asymmetry seen in a coarsely observed trajectory is not always a reliable sign of microscopic dissipation, and that the distinction between lumping (merging many microscopic states into one observed state) and milestoning (recording only crossings of localized milestone surfaces) decides which inference methods are valid. For lumped observables, the informatic entropy production of the projection never exceeds that of the full dynamics, so a nonzero time-reversal asymmetry in lumped data does imply a nonequilibrium microscopic dynamics. For milestoned observables, however, milestoning and time reversal do not commute: the backward read of a milestoned trajectory is not the milestoning of the backward microscopic trajectory. The paper states that there exist examples with $\Delta S_{\mathrm{inf}}(q)>0$ while $\Delta S_{\mathrm{inf}}(x)=\Delta S_{\mathrm{tot}}(x)=0$, so apparent entropy production can be a pure artifact of the reduction; only estimators that intentionally discard waiting-time information are presented as generally safe.

What carries the argument

The load-bearing objects are the informatic entropy production $\Delta S_{\mathrm{inf}}$, defined as the Kullback-Leibler divergence between forward and time-reversed path measures; the overdamped Markovian diffusion model of Eq. (2); and the milestoning functional $F$, which maps a continuous trajectory to the sequence of milestones it crosses. For fully observed overdamped Markov dynamics, the informatic and thermodynamic entropy productions coincide, $\Delta S_{\mathrm{tot}}=\Delta S_{\mathrm{inf}}$, and for lumped observables the log-sum inequality gives $\Delta S_{\mathrm{inf}}(q)\le \Delta S_{\mathrm{inf}}(x)$, so lumped time-reversal asymmetry is a genuine nonequilibrium signal. The argument turns on the failure of this logic under milestoning: the path weight of a milestoned process is not a marginal of the microscopic path weight, because the backwards-read milestoned trajectory differs from the milestoning of the backwards microscopic trajectory. The paper identifies the saturated-order estimator of Eq. (22), evaluated on state sequences with waiting-time information intentionally disregarded, as the only generally safe way to infer a lower bound on microscopic dissipation from such data.

What would settle it

Simulate an overdamped diffusion with detailed balance in a double-well potential, for example $\varphi(x)=x^4-3x^2+x/2$ as in the paper's Fig. 2, record milestoned trajectories with milestones at $x=\pm 0.6$, and evaluate the waiting-time-based entropy estimator that includes the directional waiting-time log ratios $\psi_{k|j}(t)$ versus $\psi_{j|i}(t)$: if it returns a positive rate while the microscopic path measure is known to be time-reversal symmetric, and the rate vanishes when waiting-time contributions are discarded as in Eq. (22), the non-commutation claim is confirmed. On experimental data, threshold a detector signal from an equilibrium system at two different thresholds and look for a waiting-time asymmetry that appears at one threshold and disappears at the other.

Watch

Extended reading notes

Core claim

The paper's central claim is that milestoning and time reversal do not commute. For a milestoned observable $q_\tau = F[(x_\tau)_{0\le \tau\le t}]$, one can have $(q_{t-\tau})_{0\le \tau\le t} \neq F[(x_{t-\tau})_{0\le \tau\le t}]$, even when the underlying microscopic process is an overdamped diffusion obeying detailed balance. Consequently, applying the naive time reversal (reading the recorded trajectory backwards) to milestoned trajectories can produce a positive informatic entropy production rate even though the microscopic dynamics produces no thermodynamic entropy at all, i.e. $\Delta S_{\mathrm{inf}}(x)=\Delta S_{\mathrm{tot}}(x)=0$. This effect, called kinetic hysteresis, appears already for first-order semi-Markov processes arising as milestonings of thermodynamically consistent overdamped diffusions and extends to milestonings of Markov jump dynamics and of non-Markovian lumped dynamics. The paper concludes that time-antisymmetric correlation functions, thermodynamic uncertainty relations, and speed limits, which are valid for lumped observables, are not generally valid for milestoned observables, and that the safe route is a saturated-order semi-Markov estimator that deliberately ignores waiting times.

Load-bearing premise

The counterexamples and the claimed safety of the saturated-order estimator assume the microscopic dynamics is an overdamped Markovian diffusion with additive noise, symmetric positive-definite diffusion proportional to temperature, and local detailed balance, as stated in Section II and used throughout Section III; if the real system is underdamped, has hidden slow degrees of freedom, or has additional sources of dissipation not expressible through the drift in Eq. (2), the specific conclusions may not carry over.

Editorial extensions

If this is right

  • Any inference method that reads a milestoned trajectory backwards and interprets the resulting asymmetry as dissipation can report $\Delta S_{\mathrm{inf}}(q)>0$ for a system at equilibrium; the false signal is caused by the coarse graining, not by the physics.
  • For lumped observables the chain $\Delta S_{\mathrm{inf}}(q)\le \Delta S_{\mathrm{inf}}(x)=\Delta S_{\mathrm{tot}}(x)$ remains valid, so detecting nonequilibrium from time-antisymmetric observables of lumped data is on safe ground.
  • Estimators of entropy production from milestoned data must either use additional structural knowledge, such as Markovian transition states or transition-path times, or use saturated-order estimators that discard waiting-time contributions.
  • Experiments with finite resolution, thresholding, or detector blind spots effectively milestone every trajectory, so claims of broken detailed balance must justify the assumed coarse-graining procedure before they can be interpreted thermodynamically.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Implicit in the paper: real-world detectors that threshold or clip signals are a form of milestoning, so a nonzero waiting-time asymmetry in a single-molecule trace is not by itself evidence of dissipation; varying the detection threshold on the same data and watching the inferred entropy production shift would test this.
  • The non-commutation result suggests that memory kernels extracted from projected equilibrium trajectories cannot be classified as even or odd under time reversal without additional assumptions, which may help reconcile conflicting reports about non-Markovian entropy production.
  • A concrete extension the paper leaves open is an analogue of the saturated-order estimator for underdamped microscopic dynamics, where velocities have definite odd time-reversal parity; such an estimator could show whether the false-signal effect weakens or disappears.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript is a perspective on how coarse-graining affects the inference of time-reversal symmetry and entropy production from observed trajectories. It contrasts lumping (deterministic many-to-one maps) with milestoning (state changes triggered by crossing hypersurfaces), arguing that only milestoning can produce Markovian coarse-grained kinetics, yet milestoning breaks the commutativity between time reversal and coarse-graining. The paper shows, citing published work, that a milestoned equilibrium system can exhibit positive informatic entropy production under the naive backward reading of the trajectory, and that lower bounds and antisymmetric-observable tests valid for lumped observables can fail for milestoned ones. It proposes saturated-order estimators (Eq. 22) as the only generally reliable inference route and singles out unproved assumptions in the literature.

Significance. If the central assertions are correct, the paper delivers an important cautionary message for experimental inference of dissipation: apparent time-reversal asymmetry in coarse-grained signals is not proof of microscopic broken detailed balance, and the type of coarse-graining is decisive. The mathematical framework for overdamped Markovian dynamics, the log-sum inequality bound, and the equivalence of informatic and thermodynamic entropy production are standard and correctly presented. The milestoning counterexamples are anchored in the authors' prior peer-reviewed work, which is a legitimate basis for a perspective. The main weakness is that the practical recommendation built around Eq. (22) is contradicted by the paper's own Outlook; as written, that advisory claim is not internally consistent. The perspective nature does not excuse unresolved tension in the central constructive message, but the issue is localized and fixable.

major comments (2)
  1. [III.B.1, Eq. (22), and IV (Outlook)] There is a direct internal contradiction about the status of the saturated-order estimator. Section III.B.1 states that if (qτ) is an n-th order semi-Markov process then Ṡ_est^{n+k} = Ṡ_est^n for all k>0 and Ṡ_est^n ≤ Ṡ_tot, citing Ref. [113]. The Outlook then states that 'rigorous results are only available for renewal dynamics' and that the thermodynamic entropy production rate 'for general n ≥ 2 semi-Markov processes was never proved rigorously.' These statements cannot both be true. Because §III.B.2 instructs readers to use saturated-order estimators in Eq. (22) as 'the presumably only (bulletproof) thing to do in general,' this inconsistency is load-bearing: the recommended remedy is not supported by the manuscript's own account for n≥2. Please specify which statement is authoritative, and if the theorem remains unproved, weaken the 'bulletproof' wording and clearly mark the n≥2 property as conjectural.
  2. [III.B.2] The claim that experimental imperfections such as finite resolution, detector blind spots, signal intensity modulation, and thresholding 'behave effectively as milestoning' is asserted without a formal or even worked-out justification. This assertion carries the practical relevance of the no-go message, since the milestoning functional is defined as a state change triggered by crossing a specified hypersurface, which is not the same as deleting or blurring parts of the trajectory. Please either prove or illustrate this equivalence with concrete examples, or state it explicitly as a conjecture and discuss the differences between milestoning and missing-data/blurring coarse-graining.
minor comments (5)
  1. [III.B.2] The sentence containing 'implies implies' should be corrected to a single 'implies'.
  2. [III.B.1, Eq. (22)] The notation \hatγ_k^{(f)} is introduced informally; define it explicitly as the final state of the subsequence before using it in the log ratio.
  3. [Fig. 2 and caption] The trajectory panels are labeled (a)-(h), but the text refers to '(a, e)', '(b-d)', and '(f-h)'; please make the panel references consistent and add exact landmarks for the lumping boundary and milestone positions.
  4. [References] Several of the works on which the central claims rest are arXiv preprints (notably Refs. [113] and, in part, [119]); provide published versions or explicit status notes where available, since the manuscript attributes a theorem to Ref. [113] that the Outlook later disavows.
  5. [Appendix A] The appendix could benefit from a one-sentence summary of the Wong-Zakai discretization point for readers unfamiliar with Refs. [130-132], to make the uniqueness argument more self-contained.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; central claims are cited to peer-reviewed prior counterexamples, with an internal rigor caveat about n>=2 semi-Markov estimators.

full rationale

No circular step meeting the required standard was found. The paper is a perspective that assembles and interprets existing results; the central assertion that milestoning and time reversal do not commute, with counterexamples having Delta_Sinf(q) > 0 while Delta_Sinf(x) = Delta_Stot(x) = 0, is not re-derived in this manuscript but cited to Refs. [62,64,67,119]. These references overlap with the present authors, but they are peer-reviewed works containing explicit counterexamples with stated assumptions (overdamped Markovian dynamics, milestoning functional, local detailed balance), so the citation is independent support rather than a reduction to the present paper's own inputs. The estimator in Eq. (22) is defined directly from observed path probabilities and is not fitted to Delta_Stot; the claimed saturation and inequality properties for n-th order semi-Markov processes are mathematical properties of that estimator, not consequences of the paper's conclusions. I do flag one self-consistency caveat that is a correctness risk rather than circularity: Section III.B.1 asserts, via Ref. [113], that for n-th order semi-Markov processes S_est^{n+k} = S_est^n and S_est^n <= S_tot, while the Outlook (Section IV) states 'rigorous results are only available for renewal dynamics. The results for the thermodynamic entropy production rate for general n >= 2 semi-Markov processes was never proved rigorously.' Thus the 'presumably only bulletproof' recommendation for saturated-order estimators is stronger than the paper's own account supports, but this does not make the estimator equal to its input by construction. Score 2 reflects the prevalence of author-overlapping citations, not logical circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its load-bearing assumptions are the standard framework of overdamped Markovian stochastic thermodynamics and the specific characterization of milestoning as a non-commuting projection. The only ad hoc element is the extrapolation from mathematical counterexamples to real experimental signals being 'already milestoned.'

free parameters (2)
  • Double-well potential coefficients (x^4 - 3x^2 + x/2) = x^4 - 3x^2 + x/2
    Used only in the illustrative simulation of Fig. 2. The central claim does not depend on these specific values, so they are not load-bearing free parameters.
  • Milestone positions A = -0.6, B = 0.6 = -0.6 and 0.6
    Chosen by hand for the pedagogical example in Fig. 2. The qualitative result of non-exponential waiting times after lumping and exponential after milestoning does not depend on the exact positions.
assumptions (4)
  • domain assumption Microscopic dynamics is an overdamped Markovian diffusion with additive noise, symmetric positive-definite diffusion matrix D proportional to T, and local detailed balance.
    Stated in Section II, Eq. (2), and used throughout the discussion of entropy production and coarse graining.
  • domain assumption For overdamped Markovian dynamics, the correct time reversal is reading the trajectory backwards, i.e., (theta x_tau) = x_{t-tau}, and thermodynamics requires equivalence of forward and backward path measures.
    Used in Section II B to derive Delta S_inf = Delta S_tot and in Section III B to discuss the naive time reversal for coarse-grained processes.
  • standard math Standard results of stochastic calculus and path measures, including the Onsager-Machlup action (Eq. 8), the log-sum inequality, the support theorem, and the Radon-Nikodym theorem, are correct and applicable.
    Invoked in Section II B and the Appendix to establish the relationships between path probabilities and entropy production.
  • ad hoc to paper Experimental signals with finite resolution, detector blind spots, thresholding, or point-spread-function modulation behave effectively as milestoning rather than lumping.
    Asserted in the 'Why should we care?' paragraph of Section III B 2. This is used to generalize the kinetic hysteresis concern to essentially all experiments, but no proof is given.

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Cite this review

Pith. "Pith review of Perspective: Time irreversibility in systems observed at coarse resolution." pith.science (2026). https://pith.science/paper/NDYXGXHZ

@misc{pith2026241202675,
  author       = {Pith},
  title        = {Pith review of: Perspective: Time irreversibility in systems observed at coarse resolution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NDYXGXHZ}},
  note         = {Machine review of arXiv:2412.02675}
}
read the original abstract

A broken time-reversal symmetry, i.e. broken detailed balance, is central to non-equilibrium physics and is a prerequisite for life. However, it turns out to be quite challenging to unambiguously define and quantify time-reversal symmetry (and violations thereof) in practice, that is, from observations. Measurements on complex systems have a finite resolution and generally probe low-dimensional projections of the underlying dynamics, which are well known to introduce memory. In situations where many microscopic states become "lumped" onto the same observable "state" or when introducing "reaction coordinates" to reduce the dimensionality of data, signatures of a broken time-reversal symmetry in the microscopic dynamics become distorted or masked. In this perspective we highlight why in defining and discussing time-reversal symmetry, and quantifying its violations, the precise underlying assumptions on the microscopic dynamics, the coarse graining, and further reductions, are not a technical detail. These assumptions decide whether the conclusions that are drawn are physically sound or inconsistent. We summarize recent findings in the field and reflect upon key challenges.

Figures

Figures reproduced from arXiv: 2412.02675 by the authors.

Figure 1
Figure 1. FIG. 1. (a) A free-energy landscape [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Numerical evaluation of one-dimensional barrier crossing. (a, e) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.