Identifiability and generality of exact, entropy-production-preserving coarse graining of Markov jump processes (open problem of Teza and Stella, PRL 125, 110601)

Teza and Stella (Phys. Rev. Lett. 125, 110601, 2020) proved that a specific coarse graining of continuous-time Markov jump processes — decimation, i.e., algebraically eliminating hidden states from the entropy-resolved master equation while retaining the resulting nonexponential residence-time statistics — leaves the average and the full fluctuation spectrum of the entropy production invariant at stationarity, and proposed recovering the hidden rates and the genuine entropy production of partially observed nonequilibrium systems by matching empirical residence-time densities $p_r(t)$, $p_l(t)$ to analytic decimation formulas. They stated that establishing general conditions under which such identification is possible and unique 'poses a future challenge' and that extending the exact results to other models is 'an open program'. Subsequent work has addressed parts of this program: Wu and Jia (2025) built a sufficient-statistics framework for parameter recovery from lumped trajectories that assumes known topology and infinite data, with uniqueness proven only for special network families; the Seifert group developed waiting-time-based estimators that yield lower bounds and partial topology inference; Teza, Stella, and GrandPre (2025) extended the exact program to lumping and found sharp fundamental limitations. The open core — general identifiability conditions (rates and topology) for the reconstruction and the class of systems admitting exact preservation — remains unresolved.

Background

Stochastic thermodynamics describes the energetics of small systems whose dynamics is dominated by thermal fluctuations, from driven colloids to molecular motors. A standard microscopic model is a continuous-time Markov jump process: a random sequence of transitions among a finite network of states, in which state $i$ sends probability to state $j$ at a fixed rate $W_{ji}$. When each transition exchanges a well-defined heat with the environment (the local detailed balance condition), the entropy produced along a trajectory is the log-ratio sum $S=\sum_k \ln(W_{i_k i_{k-1}}/W_{i_{k-1} i_k})$, and in the long-time limit its statistics concentrate according to a large-deviation principle (atypical values of the entropy-production rate become exponentially unlikely as time grows) controlled by the scaled cumulant generating function (SCGF) $\epsilon(\lambda)$, whose symmetry $\epsilon(\lambda-1)=\epsilon(-\lambda)$ expresses the fluctuation theorem.

Experiments rarely resolve all microscopic states. The two standard coarse grainings are lumping (merging indistinguishable microstates into one observed mesostate) and decimation (discarding unobserved states entirely, keeping only transitions among the surviving ones). The observed process then carries memory: the waiting time in a mesostate is nonexponential, and the reduced description is a semi-Markov process — a sequence of visited mesostates with arbitrary residence-time densities, denoted $p_r(t)$ and $p_l(t)$ for right- and left-moving departures. Classical results showed that coarse graining generally lowers the apparent entropy production (Puglisi et al., 2010; Esposito, 2012), that even in the limit of an infinite timescale separation between hidden and observed transitions an effective memoryless description fails to capture the full production (Esposito, 2012; Bo and Celani, 2014), and that semi-Markov fluctuation theorems (Esposito and Lindenberg, 2008; Andrieux and Gaspard, 2008) account only for directly observed transitions, missing the contribution of hidden current-carrying loops. Much of the modern literature therefore pursues model-free lower bounds on entropy production from partial observations, a field called thermodynamic inference (systematically reviewed by Seifert, 2026).

Against this background, Teza and Stella (2020) found an exact alternative. Eliminating the hidden states' probabilities algebraically from the entropy-resolved master equation (the probability hierarchy decomposed by the entropy produced so far) of a driven periodic nearest-neighbor chain (a spatially periodic ring subject to a uniform driving force) — and of chains with secondary loops and of a simplified molecular-motor network — yields coarse-grained equations containing higher-order time derivatives whose long-time SCGF equals the original one: decimation preserves the average, the fluctuations, and the fluctuation theorem of the entropy production, provided the memory is kept exactly. In the complementary trajectory picture, the observed dynamics is semi-Markov with analytic residence-time densities $p_r(t)$, $p_l(t)$ that encode the hidden rates; matching these densities, extracted from an empirical time series, to the decimation formulas recovers the full hidden Markov model and hence the genuine thermodynamics, including otherwise inaccessible entropy production. The authors state that 'At general level establishing conditions under which it would be possible and unique poses a future challenge', describe extension of the exact results to other models as 'an open program', and note that the strategy rests on the assumption that an underlying Markov jump description is adequate. What remains unresolved is the actual scope of this program: the general identifiability question and the class of systems for which exact preservation holds.

Problem Statement

Establish the actual scope of exact, entropy-production-preserving coarse graining of nonequilibrium Markov jump processes and its power as a thermodynamic-inference principle, by resolving the following linked components over explicitly stated model and observation classes:

(1) Identifiability and uniqueness. Under what general conditions do the coarse-grained observables — the semi-Markov residence- and jump-time densities $p_r(t)$, $p_l(t)$ between surviving mesostates, or any equivalent set of sufficient statistics of the lumped trajectory (summaries that exhaust the information the coarse-grained data carry about the hidden dynamics) — determine the underlying Markov generator (the full matrix of transition rates $W_{ji}$) uniquely up to observable equivalence, in its rates and, when the hidden topology is not known in advance, in its transition structure? An answer must resolve three sub-deliverables. (a) Rate identifiability: the conditions — necessary and sufficient, or broad sufficient conditions with counterexample delimitation — on the coarse-grained observables for unique recovery of the rates $W_{ji}$ up to observable equivalence. (b) Topology identifiability: the analogous conditions for unique recovery of the transition structure when the hidden topology is not known in advance. (c) Entropy-production recoverability: the conditions under which the full entropy production — mean and fluctuation spectrum — is thereby recoverable. Sub-deliverables (a) and (b) are jointly mandatory, but may be replaced as a pair by an explicit characterization of the observable-equivalence classes of generators that no coarse-grained data can distinguish; sub-deliverable (c) is mandatory in either branch.

(2) Generality of exactness. Determine for which systems exact coarse graining preserves the full entropy-production statistics, beyond the demonstrated nearest-neighbor periodic chains, secondary-loop attachments, the molecular-motor prototype, and the special lumping symmetries identified by Teza, Stella, and GrandPre (2025). The characterization must cover the general class of Markov jump processes — arbitrary jump-network topologies with complex hidden-state structure, including lumping that collapses current-carrying loops — and must delimit the scope of exactness for the two families outside that class, continuous-space diffusions and discrete-state microscopic dynamics that are non-Markovian, by giving for each family a definite verdict on whether exact preservation extends to it or provably cannot extend to it, with the supporting argument or counterexample; where exact preservation fails, quantify which information (e.g., which cumulants of the entropy production) survives.

These two components form the mathematical core acknowledged by the source ('a future challenge' and 'an open program'). They jointly support one objective: determining whether exact entropy-production preservation under coarse graining is a general structural principle of nonequilibrium dynamics and whether it yields a complete inference methodology for partially observed nonequilibrium systems.

References

1. Gianluca Teza and Attilio L. Stella, 'Exact Coarse Graining Preserves Entropy Production out of Equilibrium', Physical Review Letters 125, 110601 (2020). DOI: 10.1103/PhysRevLett.125.110601, https://doi.org/10.1103/PhysRevLett.125.110601 (preprint arXiv:2003.08674, https://arxiv.org/abs/2003.08674).
2. Gianluca Teza, Attilio L. Stella, and Trevor GrandPre, 'Coarse-Graining via Lumping: Exact Calculations and Fundamental Limitations' (2025). arXiv:2512.11974, https://arxiv.org/abs/2512.11974, DOI: 10.48550/arXiv.2512.11974.
3. Bingjie Wu and Chen Jia, 'Parameter Inference and Nonequilibrium Identification for Markov Networks Based on Coarse-Grained Observations', Physical Review Letters 134, 087103 (2025). DOI: 10.1103/PhysRevLett.134.087103, https://doi.org/10.1103/PhysRevLett.134.087103 (preprint arXiv:2406.19586, https://arxiv.org/abs/2406.19586).
4. Jann van der Meer, Benjamin Ertel, and Udo Seifert, 'Thermodynamic Inference in Partially Accessible Markov Networks: A Unifying Perspective from Transition-Based Waiting Time Distributions', Physical Review X 12, 031025 (2022). DOI: 10.1103/PhysRevX.12.031025, https://doi.org/10.1103/PhysRevX.12.031025 (preprint arXiv:2203.12020, https://arxiv.org/abs/2203.12020).
5. Pedro E. Harunari, Annwesha Dutta, Matteo Polettini, and Édgar Roldán, 'What to Learn from a Few Visible Transitions’ Statistics?', Physical Review X 12, 041026 (2022). DOI: 10.1103/PhysRevX.12.041026, https://doi.org/10.1103/PhysRevX.12.041026 (preprint arXiv:2203.07427, https://arxiv.org/abs/2203.07427).
6. Benjamin Ertel and Udo Seifert, 'Estimator of Entropy Production for Partially Accessible Markov Networks Based on the Observation of Blurred Transitions', Physical Review E 109, 054109 (2024). DOI: 10.1103/PhysRevE.109.054109, https://doi.org/10.1103/PhysRevE.109.054109 (preprint arXiv:2312.08246, https://arxiv.org/abs/2312.08246).
7. Alexander M. Maier, Julius Degünther, Jann van der Meer, and Udo Seifert, 'Inferring Kinetics and Entropy Production from Observable Transitions in Partially Accessible, Periodically Driven Markov Networks', Journal of Statistical Physics 191, 104 (2024). DOI: 10.1007/s10955-024-03315-7, https://doi.org/10.1007/s10955-024-03315-7 (preprint arXiv:2406.12657, https://arxiv.org/abs/2406.12657).
8. Alexander M. Maier, Udo Seifert, and Jann van der Meer, 'From Observed Transitions to Hidden Paths in Markov Networks', Physical Review Research 7, 033067 (2025). DOI: 10.1103/p4k1-1dvb, https://doi.org/10.1103/p4k1-1dvb (preprint arXiv:2504.16015, https://arxiv.org/abs/2504.16015).
9. Alexander M. Maier, Jonas H. Fritz, and Udo Seifert, 'Pedestrian’s Approach to Large Deviations in Semi-Markov Processes with an Application to Entropy Production', Physical Review E 113, 014119 (2026). DOI: 10.1103/44r5-fjdm, https://doi.org/10.1103/44r5-fjdm (preprint arXiv:2509.15077, https://arxiv.org/abs/2509.15077).
10. Jonas H. Fritz and Udo Seifert, 'Waiting-Time-Based Entropy Estimators in Continuous Space Without Markovian Events', Physical Review E 114, 034109 (2026). DOI: 10.1103/zj2b-w93z, https://doi.org/10.1103/zj2b-w93z (preprint arXiv:2603.05170, https://arxiv.org/abs/2603.05170).
11. Udo Seifert, 'Universal Bounds on Entropy Production from Fluctuating Coarse-Grained Trajectories', Nature Reviews Physics 8, 493–507 (2026). DOI: 10.1038/s42254-026-00954-5, https://doi.org/10.1038/s42254-026-00954-5 (preprint arXiv:2512.07772, https://arxiv.org/abs/2512.07772).
12. David Hartich and Aljaž Godec, 'Emergent Memory and Kinetic Hysteresis in Strongly Driven Networks', Physical Review X 11, 041047 (2021). DOI: 10.1103/PhysRevX.11.041047, https://doi.org/10.1103/PhysRevX.11.041047 (preprint arXiv:2011.04628, https://arxiv.org/abs/2011.04628).
13. Xizhu Zhao, David Hartich, and Aljaž Godec, 'Emergence of Memory in Equilibrium versus Nonequilibrium Systems', Physical Review Letters 132, 147101 (2024). DOI: 10.1103/PhysRevLett.132.147101, https://doi.org/10.1103/PhysRevLett.132.147101 (preprint arXiv:2311.12788, https://arxiv.org/abs/2311.12788).
14. David Hartich and Aljaž Godec, 'Violation of Local Detailed Balance upon Lumping Despite a Clear Timescale Separation', Physical Review Research 5, L032017 (2023). DOI: 10.1103/PhysRevResearch.5.L032017, https://doi.org/10.1103/PhysRevResearch.5.L032017 (preprint arXiv:2111.14734, https://arxiv.org/abs/2111.14734).
15. Kristian Blom, Kevin Song, Etienne Vouga, Aljaž Godec, and Dmitrii E. Makarov, 'Milestoning Estimators of Dissipation in Systems Observed at a Coarse Resolution', PNAS 121, e2318333121 (2024). DOI: 10.1073/pnas.2318333121, https://doi.org/10.1073/pnas.2318333121 (preprint arXiv:2310.06833, https://arxiv.org/abs/2310.06833).
16. Pedro E. Harunari, 'Uncovering Nonequilibrium from Unresolved Events', Physical Review E 110, 024122 (2024). DOI: 10.1103/PhysRevE.110.024122, https://doi.org/10.1103/PhysRevE.110.024122 (preprint arXiv:2402.00837, https://arxiv.org/abs/2402.00837).
17. Ramón Nartallo-Kaluarachchi, Renaud Lambiotte, and Alain Goriely, 'Coarse-Graining Nonequilibrium Diffusions with Markov Chains', Journal of Statistical Mechanics (2026) 033205. DOI: 10.1088/1742-5468/ae4f7d, https://doi.org/10.1088/1742-5468/ae4f7d (preprint arXiv:2511.05366, https://arxiv.org/abs/2511.05366).
18. Xuyan Xiang, Jieming Zhou, and Beilin Xiang, 'Moment-Based Identification of Diffusion Generators via First Hitting Times', Mathematics 14, 2586 (2026). DOI: 10.3390/math14142586, https://doi.org/10.3390/math14142586.
19. A. Puglisi, S. Pigolotti, L. Rondoni, and A. Vulpiani, 'Entropy Production and Coarse Graining in Markov Processes', Journal of Statistical Mechanics (2010) P05015. DOI: 10.1088/1742-5468/2010/05/P05015, https://doi.org/10.1088/1742-5468/2010/05/P05015.
20. Massimiliano Esposito, 'Stochastic Thermodynamics under Coarse Graining', Physical Review E 85, 041125 (2012). DOI: 10.1103/PhysRevE.85.041125, https://doi.org/10.1103/PhysRevE.85.041125 (preprint arXiv:1112.5410, https://arxiv.org/abs/1112.5410).
21. Stefano Bo and Antonio Celani, 'Entropy Production in Stochastic Systems with Fast and Slow Time-Scales', Journal of Statistical Physics 154, 1325–1351 (2014). DOI: 10.1007/s10955-014-0922-1, https://doi.org/10.1007/s10955-014-0922-1.
22. Massimiliano Esposito and Katja Lindenberg, 'Continuous-Time Random Walk for Open Systems: Fluctuation Theorems and Counting Statistics', Physical Review E 77, 051119 (2008). DOI: 10.1103/PhysRevE.77.051119, https://doi.org/10.1103/PhysRevE.77.051119.
23. David Andrieux and Pierre Gaspard, 'The Fluctuation Theorem for Currents in Semi-Markov Processes', Journal of Statistical Mechanics (2008) P11007. DOI: 10.1088/1742-5468/2008/11/P11007, https://doi.org/10.1088/1742-5468/2008/11/P11007.
24. Trevor GrandPre, Gianluca Teza, and William Bialek, 'Direct Estimates of Irreversibility from Time Series' (2024). arXiv:2412.19772, https://arxiv.org/abs/2412.19772.