Waiting-time distributions (WTDs) between observed transitions are a standard input for inferring entropy production from partially observed time series. Martínez, Bisker, Horowitz and Parrondo (2019) proposed a WTD-based estimator, S˙WTD\dot S_{\mathrm{WTD}}, claimed to detect broken detailed balance even when all observable currents vanish. Hartich and Godec showed by an explicit equilibrium molecular-motor network with a hidden enzymatic intermediate that S˙WTD≠0\dot S_{\mathrm{WTD}}\neq0 can be a false positive: the coarse-grained observation is a semi-Markov process with direction-dependent waiting-time densities, because the coarse-graining does not commute with time reversal (kinetic hysteresis, caused by non-instantaneous transition paths through hidden states). The subsequent Reply identified coarse-graining that commutes with time reversal as a sufficient condition for consistency and asserted that for first-order semi-Markov processes from decimation the WTD term vanishes whenever the affinity term does. The surviving open core is the Comment's own question: whether a microscopic model exists that upon coarse-graining yields a semi-Markov process — explicitly referring to (first-order) semi-Markov processes and not to possible generalizations to second- or higher-order semi-Markov processes — and concurrently allows one to infer broken detailed balance according to S˙WTD\dot S_{\mathrm{WTD}}, i.e., whether S˙WTD≠0\dot S_{\mathrm{WTD}}\neq0 is possible at all without kinetic hysteresis. An explicit model answering this existence question, or a no-go argument covering that class, directly and completely resolves it.

Background

In stochastic thermodynamics, mesoscopic dynamics are modeled as continuous-time Markov jump processes between microstates with transition rates kαβk_{\alpha\beta}. If the stationary distribution satisfies detailed balance, παkαβ=πβkβα\pi_\alpha k_{\alpha\beta}=\pi_\beta k_{\beta\alpha} for every pair of states, the system is at thermodynamic equilibrium; otherwise it reaches a nonequilibrium steady state with a positive mean entropy-production rate. Experiments rarely resolve all microstates: some states or transitions are hidden, and one observes a coarse-graining of the microscopic trajectory, obtained for example by lumping (merging several microstates into one observed state) or by decimation (deleting chosen states from the record). After coarse-graining the observed dynamics is generally non-Markovian, and a common reduced description is the semi-Markov process (sMP): the system jumps between observed states, with the waiting time in each state drawn from a distribution that may depend on the current state (first order) or on both the preceding and the following states (second order). Waiting-time distributions (WTDs) between observed events are directly measurable from time series.

Because the entropy production equals the Kullback–Leibler divergence (KLD) between a microscopic trajectory and its time reversal, and because coarse-graining cannot increase the KLD (the data-processing inequality), the KLD between an observed coarse-grained trajectory and its time reversal should yield a lower bound on the true entropy production. Martínez, Bisker, Horowitz and Parrondo (2019) made this operational: for semi-Markov observations they derived a compact estimator S˙WTD\dot S_{\mathrm{WTD}}, a non-negative weighted sum of KLDs between the waiting-time densities of different transitions, and claimed it reveals broken detailed balance even in the absence of observable currents (at stalling, where every measured current vanishes).

Hartich and Godec showed this claim is false as stated. Their Comment (preprint 2021, published 2024) constructs an equilibrium Markov network — a molecular motor stepping in two directions, one of which proceeds through a hidden enzymatic intermediate so that the transition path between the observed states has a finite duration — whose last-visited-observed-state process is a semi-Markov process with direction-dependent waiting-time densities, ψx(t)=(4/3)[2e−2t−2e−8t]\psi_x(t)=(4/3)[2e^{-2t}-2e^{-8t}] versus ψy(t)=(4/3)[e−2t+2e−8t]\psi_y(t)=(4/3)[e^{-2t}+2e^{-8t}] with DKL[ψx∥ψy]≈0.28D_{\mathrm{KL}}[\psi_x\|\psi_y]\approx0.28, so that S˙WTD≈1.5>0\dot S_{\mathrm{WTD}}\approx1.5>0 at full detailed balance: a false positive. The mechanism is that the coarse-graining does not commute with time reversal — reversing the observed trajectory is not the same as observing the reversed microscopic trajectory. Hartich and Godec call this non-commutation, caused by non-instantaneous transition paths over hidden states, kinetic hysteresis, the overdamped analogue of the odd parity of momenta. The same phenomenon underlies an older result of Wang and Qian (2007): for semi-Markov processes, detailed balance is necessary but not sufficient for time reversibility.

The Reply of Bisker, Martínez, Horowitz and Parrondo (2024) concedes the counterexample is essentially correct and locates the gap: the KLD expressions are exact, but the lower-bound property requires the coarse-graining to commute with time reversal, which their decimation had implicitly assumed to be local in time. The Reply also asserts that if decimation of a Markov chain yields a first-order sMP, the WTD contribution vanishes whenever the affinity contribution does, so that detecting dissipation at zero currents is possible only for second-order sMPs. In parallel, van der Meer, Ertel and Seifert (2022) and Harunari, Dutta, Polettini and Roldán (2022) independently developed a transition-based reformulation (milestoning, where observations are defined by the crossing of hypersurfaces between states, so that observed events are transitions rather than states): there the estimator is built with a modified time-reversal operation that reverses transitions and accounts for kinetic hysteresis, provably vanishes at equilibrium, and still detects nonequilibrium at stalling. Seifert's review of coarse-grained entropy-production bounds (preprint December 2025, published 2026) consolidates the field's stance: every such bound 'crucially requires' that coarse-graining and time reversal commute, and it cites the Martínez–Hartich–Godec–Bisker exchange as the instructive discussion of what happens otherwise.

What remains missing, and what the Comment's open questions target, is whether such inference can ever be genuine: whether a microscopic model exists that upon coarse-graining yields a semi-Markov process — explicitly a first-order sMP, not a generalization to second- or higher order — and concurrently allows one to infer broken detailed balance according to Martínez et al.; equivalently, since the Reply's contraction argument shows kinetic hysteresis is necessary for equilibrium false positives, whether non-vanishing waiting-time entropy production is possible at all without kinetic hysteresis. The Comment's authors state that they are not aware of any example without kinetic hysteresis that would allow a non-vanishing waiting-time entropy production.

Problem Statement

Objective: determine whether a microscopic model exists that upon coarse-graining yields a semi-Markov process — the question explicitly refers to (first-order) semi-Markov processes and not to possible generalizations to second- or higher-order semi-Markov processes — and concurrently allows one to infer broken detailed balance according to the waiting-time entropy production S˙WTD\dot S_{\mathrm{WTD}} computed with the observer's conventional time reversal; equivalently, whether S˙WTD≠0\dot S_{\mathrm{WTD}}\neq0 is possible at all without kinetic hysteresis (non-commutation of coarse-graining and time reversal caused by non-instantaneous transition paths over hidden states). An explicit model answering the existence question, or a no-go argument covering that class, directly and completely resolves the problem.

Relevant known partial results, as background rather than required steps: coarse-graining that commutes with time reversal is sufficient for S˙WTD\dot S_{\mathrm{WTD}} to vanish whenever the microscopic dynamics satisfy detailed balance and to remain a lower bound on the true entropy-production rate otherwise; non-commutation is necessary for equilibrium false positives; and the published Reply asserts — without a published proof — that for decimations yielding first-order semi-Markov processes S˙aff=0\dot S_{\mathrm{aff}}=0 implies S˙WTD=0\dot S_{\mathrm{WTD}}=0, restricting its stalling-detection claim to second-order sMPs. That assertion is a partial no-go covering only decimation at zero affinity, not the full class of models that coarse-grain to a first-order sMP, so it does not settle the existence question. A related methodological direction that the Comment motivates but does not pose as a formal question is criteria decidable from observed trajectories alone that distinguish hidden-state-induced kinetic path asymmetry from genuine thermodynamic driving; this direction is not part of the existence question above.

References

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