For a finite-state Markov jump network with periodically time-dependent transition rates satisfying local detailed balance, an external observer who sees only a subset of the transitions can estimate the system's entropy production from the statistics of waiting times between observed transitions. Maier, Degünther, van der Meer and Seifert (J. Stat. Phys. 191, 104 (2024)) proved that two such estimators are lower bounds on the mean entropy production rate in the periodic stationary state, but for their practically most convenient estimator ⟨σ^Ψ⟩pss\langle\hat\sigma_\Psi\rangle_{\rm pss} — built from waiting-time distributions marginalized over the phase of the driving, so that neither the driving phase nor a time-reversed experiment needs to be tracked — they could only conjecture the bound ⟨σ⟩pss≥⟨σ^Ψ⟩pss≥0\langle\sigma\rangle_{\rm pss}\ge\langle\hat\sigma_\Psi\rangle_{\rm pss}\ge 0, supported by numerical evidence on more than 10510^5 random networks. The open problem is to prove or disprove this inequality for arbitrary periodically driven, locally detailed balanced Markov networks. An independent line of work (Harunari, Fiore and Barato, Phys. Rev. E 110, 064126 (2024)) could prove only a weaker bound carrying a non-negative correction term requiring reversed-protocol data, and a 2026 review by Seifert still presents the periodic case as investigated but unresolved.

Background

Stochastic thermodynamics describes small systems far from equilibrium whose relevant degrees of freedom undergo a memoryless (Markovian) jump dynamics on a finite network of states: transitions from state kk to state ll occur with rates kkl(t)k_{kl}(t), and thermodynamic consistency is imposed by the local detailed balance condition ln⁡[kkl(t)/klk(t)]=Fk(t)−Fl(t)+fkl(t)\ln[k_{kl}(t)/k_{lk}(t)]=F_k(t)-F_l(t)+f_{kl}(t), where Fk(t)F_k(t) are free energies (in units of thermal energy) and fkl(t)=−flk(t)f_{kl}(t)=-f_{lk}(t) are non-conservative driving forces. When the rates are periodically driven with period T\mathcal T, the state probabilities relax to a periodic stationary state (PSS) pkpss(t)=pkpss(t+T)p^{\rm pss}_k(t)=p^{\rm pss}_k(t+\mathcal T), generalizing the nonequilibrium steady state (NESS) of time-independent driving. The mean entropy production rate in the PSS is ⟨σ⟩pss=(1/T)∫0T∑klpkpss(t)kkl(t)ln⁡pkpss(t)kkl(t)plpss(t)klk(t) dt\langle\sigma\rangle_{\rm pss}=(1/\mathcal T)\int_0^{\mathcal T}\sum_{kl}p^{\rm pss}_k(t)k_{kl}(t)\ln\frac{p^{\rm pss}_k(t)k_{kl}(t)}{p^{\rm pss}_l(t)k_{lk}(t)}\,dt, the system's total dissipation. In realistic partially accessible settings, all states are hidden and only some transitions are observable; recovering thermodynamic information from such restricted data is the field of thermodynamic inference. For an observed transition I=(ij)I=(ij) followed by the next observable transition JJ, the waiting-time distribution ψI→J(t∣t0)\psi_{I\to J}(t|t_0) gives the probability density that JJ occurs a time tt after II occurred at phase t0∈[0,T)t_0\in[0,\mathcal T) within the period; in a PSS these distributions genuinely depend on the phase t0t_0. Phase-marginalized distributions are defined as ΨI→J(t)=∫0Tppss(t0∣I)ψI→J(t∣t0) dt0\Psi_{I\to J}(t)=\int_0^{\mathcal T}p^{\rm pss}(t_0|I)\psi_{I\to J}(t|t_0)\,dt_0, where ppss(t0∣I)p^{\rm pss}(t_0|I) is the conditional probability that transition II is observed at phase t0t_0; these marginalized statistics are obtainable from measured waiting times without knowing the phase, e.g. by discarding successive waiting times so that their phases are uncorrelated. The time-reversed process (marked with tildes) has reversed rates k~ij(t)=kij(T−t)\tilde k_{ij}(t)=k_{ij}(\mathcal T-t) and reversed transitions I~\tilde I; its own phase-marginalized waiting-time distributions ΨJ~→I~(t)\Psi_{\tilde J\to\tilde I}(t) enter the estimator below. For NESSs, estimating dissipation from waiting times has a proven foundation: Roldán and Parrondo (2010) showed that a Kullback–Leibler divergence (relative entropy) between forward and backward waiting-time distributions bounds dissipation, Skinner and Dunkel (2021) developed waiting-time estimators further, and van der Meer, Ertel and Seifert (Phys. Rev. X 12, 031025 (2022)) proved that for a NESS the transition-based estimator σWTD=∑I,JνI∫0∞ψI→J(t)ln⁡ψI→J(t)ψJ~→I~(t) dt\sigma_{\rm WTD}=\sum_{I,J}\nu_I\int_0^\infty\psi_{I\to J}(t)\ln\frac{\psi_{I\to J}(t)}{\psi_{\tilde J\to\tilde I}(t)}\,dt is a non-negative lower bound on the entropy production rate. Maier, Degünther, van der Meer and Seifert (J. Stat. Phys. 191, 104 (2024)) extended this program to periodically driven networks. They introduced three estimators: (a) ⟨σ^ψ⟩pss\langle\hat\sigma_\psi\rangle_{\rm pss}, built from the phase-resolved distributions ψI→J(t∣t0)\psi_{I\to J}(t|t_0) and their reversed counterparts, which they proved to be a lower bound via a trajectory-level fluctuation relation, but which requires waiting-time data resolved by phase and, in general, data from the time-reversed driving protocol; (b) the phase-marginalized estimator ⟨σ^Ψ⟩pss=∑I,J⟨nI⟩pss∫0∞ΨI→J(t)ln⁡ΨI→J(t)ΨJ~→I~(t) dt\langle\hat\sigma_\Psi\rangle_{\rm pss}=\sum_{I,J}\langle n_I\rangle_{\rm pss}\int_0^\infty\Psi_{I\to J}(t)\ln\frac{\Psi_{I\to J}(t)}{\Psi_{\tilde J\to\tilde I}(t)}\,dt, where ⟨nI⟩pss\langle n_I\rangle_{\rm pss} is the mean rate of observed transitions II averaged over one period; and (c) ⟨σ^pk⟩pss\langle\hat\sigma_{pk}\rangle_{\rm pss}, assembled from boundary-state probabilities and rates that the same paper shows how to infer, also proved to be a lower bound but often looser, with no fixed ordering against (b). In a NESS, estimator (a) reduces to (b), so the NESS case of (b) is proven; the conjecture is therefore genuinely a periodic-driving question. Since ΨI→J(t)\Psi_{I\to J}(t) and ΨJ~→I~(t)\Psi_{\tilde J\to\tilde I}(t) are normalized waiting-time densities, the inner sum over JJ in ⟨σ^Ψ⟩pss\langle\hat\sigma_\Psi\rangle_{\rm pss} is a non-negative Kullback–Leibler divergence, so the second inequality ⟨σ^Ψ⟩pss≥0\langle\hat\sigma_\Psi\rangle_{\rm pss}\ge 0 holds automatically from the definition and the conjectural content of the chain is the first inequality alone. The difficulty is that the NESS proof factorizes coarse-grained path weights into products of waiting-time densities and applies a fluctuation relation, whereas under periodic driving the observed-transition process carries a phase coordinate whose marginalization destroys this factorization and couples the forward and reversed phase statistics in a non-standard way. Generalizations of the thermodynamic uncertainty relation to periodic driving exist but use current statistics and yield different, generally incomparable bounds. Numerically, the inequality ⟨σ⟩pss≥⟨σ^Ψ⟩pss≥0\langle\sigma\rangle_{\rm pss}\ge\langle\hat\sigma_\Psi\rangle_{\rm pss}\ge 0 held for over 10510^5 randomly chosen periodically driven networks on unicyclic (single-cycle), diamond-shaped and more complex graphs, which led the authors to state the bound as a conjecture whose proof 'remains as open theoretical challenge'.

Problem Statement

Prove or disprove the conjectured inequality

⟨σ⟩pss  ≥  ⟨σ^Ψ⟩pss  ≥  0\langle\sigma\rangle_{\rm pss}\;\ge\;\langle\hat\sigma_\Psi\rangle_{\rm pss}\;\ge\;0

for arbitrary periodically driven Markov networks. Concretely: consider a finite-state, continuous-time Markov jump network whose transition rates kij(t)=kij(t+T)k_{ij}(t)=k_{ij}(t+\mathcal T) are T\mathcal T-periodic and satisfy local detailed balance, relaxing to a periodic stationary state ppss(t)p^{\rm pss}(t); let the set of observable transitions be nonempty and, as in the source paper's standing setting, contain at least one pair of mutually reverse transitions (both directions of one edge, all states being hidden), with phase-conditioned waiting-time distributions ψI→J(t∣t0)\psi_{I\to J}(t|t_0) between successive observed transitions, phase-marginalized distributions ΨI→J(t)=∫0Tppss(t0∣I) ψI→J(t∣t0) dt0\Psi_{I\to J}(t)=\int_0^{\mathcal T}p^{\rm pss}(t_0|I)\,\psi_{I\to J}(t|t_0)\,dt_0 (and the corresponding distributions ΨJ~→I~(t)\Psi_{\tilde J\to\tilde I}(t) of the time-reversed process with rates k~ij(t)=kij(T−t)\tilde k_{ij}(t)=k_{ij}(\mathcal T-t)), and ⟨nI⟩pss\langle n_I\rangle_{\rm pss} the mean rate of observed transitions II averaged over one period; define, as in the source paper (its Eqs. (3) and (14)),

⟨σ^Ψ⟩pss  ≡  ∑I,J⟨nI⟩pss∫0∞ΨI→J(t) ln⁡ΨI→J(t)ΨJ~→I~(t) dt,\langle\hat\sigma_\Psi\rangle_{\rm pss}\;\equiv\;\sum_{I,J}\langle n_I\rangle_{\rm pss}\int_0^\infty\Psi_{I\to J}(t)\,\ln\frac{\Psi_{I\to J}(t)}{\Psi_{\tilde J\to\tilde I}(t)}\,dt,

and let ⟨σ⟩pss\langle\sigma\rangle_{\rm pss} be the mean total entropy production rate in the PSS. A complete answer is either (i) a rigorous proof of the inequality for the full class of such networks — stating explicitly any additional assumptions invoked (e.g. ergodicity or irreducibility) and confirming that the argument reduces to the known proven NESS bound when the driving is time-independent — or (ii) an explicit counterexample: a concrete network with explicitly specified graph, period, and periodic rate functions satisfying local detailed balance, together with a certified evaluation (an analytic proof, an exact/rational-arithmetic computation, or an interval-arithmetic computation with an explicit error bound smaller than the exhibited violation) showing a strict violation of ⟨σ⟩pss≥⟨σ^Ψ⟩pss\langle\sigma\rangle_{\rm pss}\ge\langle\hat\sigma_\Psi\rangle_{\rm pss}. The source paper additionally poses two satellite directions — transferring the inference scheme to continuous-space Langevin dynamics under periodic driving, and applying the phase-resolved waiting-time estimator to experimental data of periodically driven small systems with realistic statistics — which remain open as research programs but are not part of the completion boundary of this problem.

References

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