Fundamental limits of entropy-production inference from noisy single-molecule time series and the general theory of the optimal sampling interval

Single-molecule experiments record a noisy, discretely sampled observable $Y(t)$ in place of the true microscopic Markovian dynamics $X(t)$ of a nonequilibrium steady state. Rigorous model-free inference schemes then return only lower bounds on the entropy production, the gap grows with measurement noise, and, in the paradigmatic model studied, even non-Markov estimators of arbitrarily high order saturate strictly below the true value; meanwhile the dependence of the achievable accuracy on the sampling interval is non-monotonic and known only model-by-model. The problem asks for a general theory with two coupled deliverables: (i) a characterization of the fundamental limit of entropy-production estimation from noisy observations — the supremum over estimators as a function of the noise channel and temporal resolution, together with necessary and sufficient conditions under which the noise-induced gap vanishes or is irrecoverably positive; and (ii) a general theory of the optimal sampling interval (and, where applicable, observation-channel parameters) that maximizes inference accuracy, explaining within one framework the regimes where finer temporal resolution helps and where it hurts.

Background

Stochastic thermodynamics describes small systems — molecular motors, enzymes, driven colloids — whose dynamics is dominated by thermal fluctuations. A continuous-time Markov jump process $X(t)$ on a network of states with rate matrix $\mathbf{K}$ dissipates energy when it is driven out of equilibrium; the rate of dissipation is quantified by the entropy production rate, which equals the Kullback–Leibler (KL) divergence per unit time between the probability of a forward trajectory and that of its time-reversed twin, and is positive exactly when detailed balance is broken. Most interesting microscopic systems operate in a nonequilibrium steady state (NESS), where a stationary probability distribution coexists with nonzero steady probability currents around cycles.

Single-molecule and single-particle tracking experiments cannot observe $X(t)$ directly. Two experimental limitations degrade the record: finite temporal resolution (the state is probed only at intervals $\tau$, or at the random arrival times of photons) and measurement noise (the reported value $Y$ may differ from the true state $X$; the statistics of misreporting are encoded in an error matrix $\mathbf{R}$ with entries $P(Y|X)$, the discrete analogue of a point-spread function — the blur kernel with which an imaging system maps a point source to an observed spot). Gingrich, Igoshin, Kolomeisky, and Makarov (arXiv:2606.23619) analyzed thermodynamic inference under both limitations for a Markovian $X(t)$ in a NESS. Two strategies exist. Strategy 1 (deconvolution) inverts the noise: from the statistics of $Y$-trajectories one recovers the propagator $e^{\mathbf{K}\tau}$ of $X$ — the matrix of transition probabilities over a time interval $\tau$ — by inverting $\mathbf{R}$ and then obtains $\mathbf{K}$ by a matrix logarithm. This requires $\mathbf{R}$ to be known and invertible, and the matrix logarithm becomes numerically unstable when $\tau$ exceeds the inverse of some eigenvalues of $\mathbf{K}$ — the short-time dynamics is then unrecoverable from the data. Strategy 2 (direct estimation) applies entropy-production estimators to $Y(t)$ as if it were the true dynamics. Because the noise-degraded $Y(t)$ is a non-Markovian process with long memory even when $X(t)$ is Markovian, estimators of finite Markov order $k$ — which model the observed sequence as a chain whose next value depends on the preceding $k$ recorded values — can be improved systematically by raising $k$; nevertheless the KL divergence between $Y$-trajectories and their time reverses remains a rigorous lower bound on the true entropy production, and in the model studied the $k\to\infty$ limit saturates strictly below the true value, i.e. information about the arrow of time is irreversibly lost in the noise. A third, model-free alternative uses the thermodynamic uncertainty relation (TUR) — an inequality stating that the entropy production lower-bounds the squared mean over the variance of any steady-state current — applied to a current computed from the noisy record; for a unicyclic photon-emission model (a three-state motor whose states form a single driven cycle, observed only through the colors of emitted photons) the paper proves that this bound decomposes into the intrinsic TUR looseness (which vanishes only near equilibrium) plus a measurement penalty (which closes only in the ideal limit of zero error probability and infinite photon rate).

The dependence of all these estimates on the observation design is nontrivial and non-monotonic. In a 20-node ring network with noisy state readout, an optimal intermediate sampling interval exists: too small an interval and noise-induced spurious transitions dominate, too large and full rotations around the cycle are missed. In a fully connected 8-node network, increasing the sampling interval only degrades the estimate. In the photon-emission model, the estimates peak at an intermediate photon detection rate and then fall as wrong-color photon noise accumulates. Related phenomena were reported in the milestoning literature (milestoning records a trajectory only through the times at which it crosses a chosen set of marked surfaces), where observing the system at sparser milestones can improve dissipation estimates from coarse-grained data, and in studies of finite-resolution inference, where finer temporal and spatial resolution slows the convergence of finite statistics so that coarser observation can be beneficial for short records. What is missing is any general, model-independent account of these effects: a characterization of the best achievable entropy-production estimate as a function of the noise channel and resolution, and a theory that predicts the optimal sampling interval for a given system and measurement setup.

Problem Statement

Let $X(t)$ be a continuous-time Markov process (discrete-state jump network with rate matrix $\mathbf{K}$, or overdamped Langevin dynamics — diffusion of a particle in a force field, governed by a first-order stochastic differential equation) in a nonequilibrium steady state with entropy production rate $\langle\dot{S}\rangle$. The process is observed only through a noisy channel with error matrix $\mathbf{R}$ (entries $P(Y|X)$) at temporal resolution $\tau$, or through a photon-emission channel with total emission rate $\mu$ and per-emission error probability $\epsilon$; the experimenter records only the resulting process $Y(t)$, which is generally non-Markovian. Establish a general theory of thermodynamic inference from such records, consisting of two coupled deliverables:

(1) Fundamental limits. Characterize the best achievable entropy-production estimate from the noisy observable: the supremum, over all estimators computable from the statistics of $Y$ in the infinite-data limit (i.e. from the law of the observed process, including $k$-th order non-Markov KL estimators in the limit $k\to\infty$ and thermodynamic-uncertainty-relation-based estimators), of the attainable lower bound on $\langle\dot{S}\rangle$, as a function of the noise channel and the temporal resolution. In particular, determine necessary and sufficient conditions on the channel, the resolution, and the dynamics under which this supremum equals $\langle\dot{S}\rangle$ (the noise-induced information loss vanishes), as opposed to remaining strictly below it (information about the arrow of time is irrecoverably destroyed by the noise), and quantify the gap in the latter case.

(2) Optimal observation design. Derive from the limit characterization in (1) a general theory for choosing the sampling interval $\tau$ — and, where applicable, the observation-channel parameters ($\mu$, $\epsilon$) — that maximizes the accuracy of thermodynamic inference, whether measured by the tightness of the achievable lower bound or by the convergence of finite-statistics estimators. The theory must explain, within a single framework, the regimes already documented model-by-model: where an optimal intermediate sampling interval exists (noise-induced spurious transitions dominating at short intervals while dynamical events are missed at long ones), where coarser observation only degrades the estimate, and where the estimate is non-monotonic in the photon detection rate.

The theory should be stated for a well-defined class of dynamics and observation channels, proven at the level of rigor standard in stochastic thermodynamics, and consistent with the model-level results established in the source literature — the paradigmatic exactly solvable models (the 3-state noisy random walk, the unicyclic photon-emission motor, and the ring and fully connected networks of arXiv:2606.23619, whose rate matrices are publicly available) on which the documented behaviors were demonstrated. Progress that characterizes only a restricted but explicitly identified family of networks or channels counts as partial progress toward the same objective, provided the restriction is stated and its reason understood.

References

1. T. R. Gingrich, O. A. Igoshin, A. B. Kolomeisky, D. E. Makarov (2026), 'Thermodynamic inference from noisy single-molecule time series', arXiv:2606.23619, DOI 10.48550/arXiv.2606.23619, https://arxiv.org/abs/2606.23619
2. K. Song, D. E. Makarov, E. Vouga (2024), 'Information-theoretical limit on the estimates of dissipation by molecular machines using single-molecule fluorescence resonance energy transfer experiments', J. Chem. Phys. 161, 044111, DOI 10.1063/5.0218040, https://doi.org/10.1063/5.0218040
3. M. T. Bauer, U. Seifert, J. van der Meer (2025), 'Stroboscopic measurements in Markov networks: exact generator reconstruction vs. thermodynamic inference', J. Phys. A: Math. Theor. 58, 125001, DOI 10.1088/1751-8121/adbc52, https://doi.org/10.1088/1751-8121/adbc52
4. J. H. Fritz, B. Ertel, U. Seifert (2025), 'Entropy estimation for partially accessible Markov networks based on imperfect observations: Role of finite resolution and finite statistics', Phys. Rev. E 111, 044106, DOI 10.1103/physreve.111.044106, arXiv:2412.04102, https://arxiv.org/abs/2412.04102
5. K. Blom, K. Song, E. Vouga, A. Godec, D. E. Makarov (2024), 'Milestoning estimators of dissipation in systems observed at a coarse resolution', Proc. Natl. Acad. Sci. U.S.A. 121, e2318333121, DOI 10.1073/pnas.2318333121, https://doi.org/10.1073/pnas.2318333121
6. Q. Yu, P. E. Harunari (2024), 'Dissipation at limited resolutions: power law and detection of hidden dissipative scales', J. Stat. Mech. 2024, 103201, DOI 10.1088/1742-5468/ad8152, arXiv:2407.13707, https://arxiv.org/abs/2407.13707
7. P. E. Harunari, C. E. Fiore, A. C. Barato (2024), 'Inference of entropy production for periodically driven systems', Phys. Rev. E 110, 064126, DOI 10.1103/physreve.110.064126, arXiv:2406.12792, https://arxiv.org/abs/2406.12792
8. Y. Cheng, R. Bao, Z. Hou (2026), 'Hierarchical Reconstruction of Time-arrow from Multi-time Correlations', arXiv:2604.25749, DOI 10.48550/arXiv.2604.25749, https://arxiv.org/abs/2604.25749
9. A. M. Maier, J. Degünther, J. van der Meer, U. Seifert (2024), 'Inferring kinetics and entropy production from observable transitions in partially accessible, periodically driven Markov networks', arXiv:2406.12657, DOI 10.48550/arXiv.2406.12657, https://arxiv.org/abs/2406.12657
10. M. Schweiger, A. Saurabh, S. Pressé (2025), 'A cautious user's guide in applying HMMs to physical systems', J. Chem. Phys. 163, 210901, DOI 10.1063/5.0284206, https://doi.org/10.1063/5.0284206
11. U. Seifert (2025), 'Universal bounds on entropy production from fluctuating coarse-grained trajectories', arXiv:2512.07772, DOI 10.48550/arXiv.2512.07772, https://arxiv.org/abs/2512.07772
12. J. H. Fritz, U. Seifert (2026), 'Waiting-time-based entropy estimators in continuous space without Markovian events', Phys. Rev. E 114, 034109, DOI 10.1103/zj2b-w93z, arXiv:2603.05170, https://arxiv.org/abs/2603.05170
13. A. C. Barato, U. Seifert (2015), 'Thermodynamic uncertainty relation for biomolecular processes', Phys. Rev. Lett. 114, 158101, DOI 10.1103/physrevlett.114.158101, https://doi.org/10.1103/physrevlett.114.158101
14. O. A. Igoshin, A. B. Kolomeisky, D. E. Makarov (2025), 'Uncovering dissipation from coarse observables: a case study of a random walk with unobserved internal states', J. Chem. Phys. 162, 034111, DOI 10.1063/5.0247331, https://doi.org/10.1063/5.0247331
15. C. Dieball, A. Godec (2025), 'Perspective: Time irreversibility in systems observed at coarse resolution', J. Chem. Phys. 162, 090901, DOI 10.1063/5.0251089, https://doi.org/10.1063/5.0251089