Tight and valid entropy-production inference from partially observed Langevin dynamics under spatial coarse-graining

When a nonequilibrium system described by coupled overdamped Langevin equations is only partially observed, the total entropy production rate (EPR) cannot be computed directly and must be bounded from the observed data. Ghosal and Bisker (Phys. Chem. Chem. Phys. 24, 24021 (2022)) studied the extreme case in which a single observed continuous variable (the hair-bundle tip position, with the motor position hidden) is spatially coarse-grained into a few discrete states, producing a second-order semi-Markov process with no observable net current; the only irreversibility signal is the asymmetry between forward and backward waiting-time distributions, quantified by a Kullback-Leibler divergence (WTD-KLD) that lower-bounds the total EPR. They found the bound typically one to two orders of magnitude loose, sensitive to an ad hoc partition choice, statistically fragile for finite trajectories, and blind to the affinity dissipation erased when coarse-graining destroys the underlying cycles. The ensuing problem - constructing, for partially observed spatially coarse-grained Langevin dynamics, an inference framework that is provably valid (free of kinetic-hysteresis artifacts), substantially tighter, equipped with a data-driven coarse-graining prescription and finite-sample error control, and able to recover or provably bound the erased affinity contribution - remains open. Substantial 2023-2026 progress exists on each facet, but no framework closes them jointly for continuous spatial coarse-graining.

Background

Stochastic thermodynamics assigns an entropy production rate (EPR) to nonequilibrium steady states of small systems coupled to heat baths; the EPR quantifies dissipation and equals the Kullback-Leibler divergence (KLD, the relative entropy $D[p\|q]=\int p\ln(p/q)$) between the probability measure of a forward trajectory and that of its time reverse. In experiments on molecular motors, active biomaterials, and living cells, only some degrees of freedom are observed, often at finite spatiotemporal resolution, so the total EPR must be inferred (bounded) from partial data - the subject of thermodynamic inference.

For Markovian dynamics on discrete states that are partially observed, the estimator of Martínez, Bisker, Horowitz and Parrondo (Nat. Commun. 10, 3542 (2019)) splits the KLD-based EPR estimate into an affinity EPR $\mathrm{EPR}_{\mathrm{aff}}$, carried by observable net currents around cycles of the coarse-grained network, and a waiting-time EPR $\mathrm{EPR}_{\mathrm{WTD}}$, carried by differences between the probability densities of the waiting times spent in a state before forward and backward transitions. A waiting-time distribution (WTD) is the density of the time the system dwells in a coarse-grained state before the next transition; when the coarse-grained process is a second-order semi-Markov process - one whose WTDs depend on the previously visited state, as arises when adjacent microscopic states are lumped (merged into a single macrostate) - $\mathrm{EPR}_{\mathrm{WTD}}$ can be nonzero even with zero net current, so that $\mathrm{EPR}_{\mathrm{aff}}=0$.

Ghosal and Bisker (2022) applied this estimator to a paradigmatic active system: the mechanosensory hair bundle of auditory cells, described by two coupled overdamped Langevin equations for the experimentally observed tip position $X_1$ and the hidden molecular-motor position $X_2$, driven out of equilibrium by a nonconservative motor force $F_{\max}$ with calcium-mediated feedback strength $S$ and an effective-to-bath temperature ratio $T_{\mathrm{eff}}/T>1$. Coarse-graining $X_1$ into $N$ discrete spatial states (equal or unequal divisions) turns the observed trajectory into a linear chain of states - a second-order semi-Markov process with $\mathrm{EPR}_{\mathrm{aff}}=0$, because the coarse-graining erases the fundamental cycle of the underlying dynamics - so the lower bound on the total EPR comes only from the WTD asymmetry, $\mathrm{EPR}_{\mathrm{WTD}}=(1/\tau)\sum_{ijk}p(ijk)D[\Psi(t|ijk)\|\Psi(t|kji)]$, where $\Psi(t|ijk)$ is the WTD for the transition sequence $i\to j\to k$ and $\tau$ is the mean step duration. Two families of alternative bounds exist for the same setting: thermodynamic and kinetic uncertainty relations (TUR/KUR), which bound the EPR by the precision of fluctuating currents and the number of transition events, and optimization-based bounds that search over hidden dynamics consistent with observed statistics.

The paper's findings define the problem. First, the bound is loose: across the driving-parameter grid, $\mathrm{EPR}_{\mathrm{WTD}}/\mathrm{EPR}_{\mathrm{tot}}$ ranges from roughly $10^{-3}$ to $0.1244$, i.e. one to two orders of magnitude, even at the finest resolution studied ($N=7$, unequal spacing). Second, the spatial partition is critical and chosen ad hoc: equal-spacing partitions fail for rugged trajectories because boundary states are undersampled, while unequal partitions improve tightness without any general data-driven prescription. Third, the estimator needs long trajectories for stable WTD estimation (errors decrease and plateau only with increasing simulation length), and no minimum-data guidelines, bias corrections, or noise-regularized density estimation are provided. Fourth, when coarse-graining erases the fundamental cycles ($\mathrm{EPR}_{\mathrm{aff}}=0$), the affinity contribution to dissipation is invisible to the estimator; approaching the total EPR would require additional observables (as in the two-observable TUR of Roldán et al. for the same system), instrumentally resolved hidden variables, or complementary inference.

A further subtlety, clarified after the source appeared, concerns validity: in a 2024 Comment and Reply exchange (Hartich and Godec; Bisker, Martínez, Horowitz, Parrondo), it was established that the WTD-KLD can be positive even at equilibrium when the coarse-graining does not commute with time reversal - so-called kinetic hysteresis, arising when hidden states make transition paths non-instantaneous - so guaranteed validity of waiting-time-based bounds requires explicit commutation conditions on the coarse-graining map. For spatial partitions of overdamped Langevin variables, where transitions between regions have intrinsically finite duration, these conditions are delicate and were handled only later within specific schemes that define transition events as crossings of interior hypersurfaces (so-called milestones) placed between the spatial regions, or as other sharply specified events, rather than by region occupancy alone.

Problem Statement

For a nonequilibrium system governed by coupled overdamped Langevin equations in which only a subset of variables is observed - paradigmatically, the hair-bundle tip position $X_1(t)$ with the motor position hidden - and in which the observed continuous trajectory is spatially coarse-grained into finitely many states (so that the observed process is a second-order semi-Markov process and, in the regime central to the source, all observable net currents vanish and the affinity contribution $\mathrm{EPR}_{\mathrm{aff}}$ is erased), construct a complete inference framework for the total entropy production rate that simultaneously achieves:

1. Validity: provable conditions on the spatial coarse-graining (commutation with time reversal, absence of kinetic-hysteresis artifacts) under which the waiting-time-based estimate, or a successor estimator, is a guaranteed lower bound on the total EPR for coarse-grained continuous Langevin variables.
2. Tightness: a bound that is provably tighter than the plain WTD-KLD of the source on the sub-class on which that plain bound is itself a valid lower bound, namely the coarse-grainings and event definitions satisfying the conditions of component 1 (the kinetic-hysteresis regime of non-commuting spatial partitions lies outside this comparison, since there the plain WTD-KLD is not a lower bound and no lower bound can dominate it, and is governed by component 1; a proven domination is required on the comparison sub-class, not tightness demonstrated only on the source's driving-parameter grid or on another single instance) - ideally together with a characterization of the best achievable bound given the observed coarse-grained statistics, rather than only empirical tightness on particular systems.
3. Optimal coarse-graining: a general, data-driven prescription for the spatial partition (placement and number of states) that is statistically optimal for the available trajectory length in a named sense - among partitions that can be prescribed from the observed coarse-grained statistics alone, the one that minimizes the mean-squared error of the resulting total-EPR estimate at that trajectory length - replacing the ad hoc equal/unequal divisions of the source.
4. Finite-sample reliability: quantitative error control for the estimator from finite trajectories - bias correction, regularized or nonparametric density estimation robust to measurement noise - with practical minimum-data guidance.
5. Erased affinity: a characterization of when the affinity dissipation destroyed by cycle-erasing coarse-graining can be recovered or bounded from the observed data alone, or a proof that it cannot (an identifiability limitation), including how additional observables or partially resolved hidden variables should be combined with waiting-time information.

These components jointly support one objective - reliable quantification of dissipation from partially observed, spatially coarse-grained Langevin data - and are to be understood as facets of that single objective, not as separable deliverables. The problem is complete when such a framework, together with the guarantees requested in components 1-5, is established for this class of systems; the active hair-bundle model and analogous partially observed Langevin systems serve as the paradigmatic test class. Each component must be discharged at that class level in the stated strength: component 1 as explicit conditions together with a proof that the waiting-time-based or successor estimate is a guaranteed lower bound under them, component 2 as the proven domination over the plain WTD-KLD bound on the comparison sub-class described there, component 3 as the stated optimality property at the available trajectory length, component 4 as explicit finite-trajectory error control (an error bound or confidence interval) together with minimum-data guidance, and component 5 as a proved recovery or bounding construction, or a proved impossibility; closing a proper subset of the five does not resolve the problem. No restriction to particular parameter values, geometries, or model subclasses is intended.

References

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