Garilli and Frezzato (Phys. Rev. E 112, 044141 (2025)) derived, for a steady-state Markov jump process on a finite irreducible network monitored through a single bidirectional transition channel, exact expressions for the squared coefficient of variation (their 'precision coefficient') of the integrated current, and connected it to the precision coefficients of the forward and backward recurrence times — but only at the level of second moments. They posed three challenges: a formal analysis of the interrelation between the full probability distributions of the current and of the recurrence times beyond second moments; a derivation of TUR-type bounds for the finite-time channel-specific precision directly from the exact expressions; and new inequalities combining the steady-state entropy-production rate with local channel observables. Their own follow-up (Phys. Rev. E 113, 054127 (2026)) resolved the long-time part of the third challenge with the dissipation bound built from the recurrence-time precision asymmetry, and their record-times paper (arXiv:2609.09479) extended the second-moment equivalence to multicyclic networks. The open core is the beyond-second-moment counting-timing correspondence — what the recurrence-time statistics encode about the full counting statistics of the current — together with the derivation of thermodynamic bounds directly from the exact channel-resolved expressions at finite monitoring times, which remains open because the derivation in that follow-up is strictly asymptotic. A detailed analysis of the temporal asymmetries beyond second moments is a route to sharpening dissipation estimates.

Background

A continuous-time Markov jump process on a finite irreducible network of states (irreducible meaning that every state is reachable from every other state through some sequence of jumps), with time-independent jump rates, admits a unique steady state with occupation probabilities pissp^{ss}_i and probability fluxes Fij=pisski→jF_{ij}=p^{ss}_i k_{i\to j}. This is the reference model for conformational dynamics of enzymes, molecular-motor stepping, and low-copy-number chemical systems. When a single bidirectional transition channel α↔β\alpha\leftrightarrow\beta between two connected sites is monitored, two families of statistics are natural. The integrated current Nαβ(t)N_{\alpha\beta}(t) is the net number of α→β\alpha\to\beta jumps minus β→α\beta\to\alpha jumps in a window [0,t][0,t], with mean ⟨Nαβ(t)⟩=Jαβt\langle N_{\alpha\beta}(t)\rangle=J_{\alpha\beta}t where Jαβ=Fαβ−FβαJ_{\alpha\beta}=F_{\alpha\beta}-F_{\beta\alpha} is the steady-state current. The recurrence times ταβ\tau_{\alpha\beta} and τβα\tau_{\beta\alpha} are the times elapsed between successive occurrences of the α→β\alpha\to\beta and β→α\beta\to\alpha jumps respectively; before a given transition repeats, the backward one may occur several times; the recurrence time of a single transition is nonetheless measured blind to those intervening backward jumps. Each occurrence of a transition resets the system to its arrival state, so the recurrence times of α→β\alpha\to\beta and of β→α\beta\to\alpha are each independent and identically distributed, i.e., each sequence is a renewal process; it is the statistical coupling between the forward and backward recurrence sequences, not non-renewal of either sequence, that makes the beyond-second-moment counting-timing correspondence non-trivial. Their means are fixed by the fluxes, τˉαβ=1/Fαβ\bar\tau_{\alpha\beta}=1/F_{\alpha\beta}. Following Garilli and Frezzato, the precision coefficients are squared coefficients of variation (variance over squared mean): PαβN(t)=(⟨N2⟩−⟨N⟩2)/⟨N⟩2P^{\mathcal N}_{\alpha\beta}(t)=(\langle N^2\rangle-\langle N\rangle^2)/\langle N\rangle^2 for the current, and Pαβτ=(τ2‾−τˉ2)/τˉ2P^{\tau}_{\alpha\beta}=(\overline{\tau^2}-\bar\tau^2)/\bar\tau^2 for each recurrence time; in single-molecule enzymology PτP^{\tau} is called the randomness parameter, and the Fano factor Var N/⟨N⟩\mathrm{Var}\,N/\langle N\rangle equals JαβT∞J_{\alpha\beta}T_\infty at long times. Two classical bounds constrain the current's precision: the kinetic uncertainty relation (KUR) PαβN(t)≥(κt)−1P^{\mathcal N}_{\alpha\beta}(t)\ge(\kappa t)^{-1} with κ\kappa the total activity (average number of jumps per unit time), and the thermodynamic uncertainty relation (TUR) PαβN(t)≥2(σsst)−1P^{\mathcal N}_{\alpha\beta}(t)\ge 2(\sigma^{ss}t)^{-1} for networks with all-bidirectional channels, where σss\sigma^{ss} is the steady-state entropy-production rate. The TUR is loose far from equilibrium, which motivates bounds using channel-resolved information. On the timing side, an Aldous-Shepp-type theorem gives Pτ≥1/MP^{\tau}\ge 1/M for a network with MM states (equality for unicyclic networks — those whose state graph consists of a single cycle — with irreversible transitions), a standard tool for bounding the minimal number of states of a hidden mechanism from observed timing regularity. Garilli and Frezzato (Phys. Rev. E 112, 044141 (2025)) derived an exact expression for PαβN(t)P^{\mathcal N}_{\alpha\beta}(t) (their Eq. 15, via the moment-generating function) on arbitrary irreducible networks at finite time, recast it (their Eq. 20) as PαβN(t)=T∞/t+t−2∫0t(t−t′)ϕ(t′) dt′P^{\mathcal N}_{\alpha\beta}(t)=T_\infty/t+t^{-2}\int_0^t(t-t')\phi(t')\,dt', where the long-time coefficient T∞T_\infty (their Eq. 21) is built from the difference of the two recurrence-time precisions, the current, the rectifying efficiency ϵ=Jαβ/(Fαβ+Fβα)\epsilon=J_{\alpha\beta}/(F_{\alpha\beta}+F_{\beta\alpha}), and conditional sojourn times, and derived kinetic and thermodynamic inequalities linking the recurrence-time precision asymmetry to the current and to σss\sigma^{ss}. This extended to generic networks, finite times, and reversible channels the classic long-time equivalence between the randomness parameter and the Fano factor previously known for processive enzymes with an irreversible completion step (Wierenga, ten Wolde and Becker). All interrelations established in that work, however, involve only the first two moments of the distributions.

Problem Statement

Consider a continuous-time Markov jump process at steady state on a finite irreducible network of states with time-independent rates, monitored through a single bidirectional transition channel α↔β\alpha\leftrightarrow\beta. Let Nαβ(t)N_{\alpha\beta}(t) be the integrated current over [0,t][0,t], and let ταβ\tau_{\alpha\beta}, τβα\tau_{\beta\alpha} be the recurrence times of the α→β\alpha\to\beta and β→α\beta\to\alpha transitions. The established results connect these quantities only at the level of second moments (the exact finite-time expression for the precision coefficient PαβN(t)P^{\mathcal N}_{\alpha\beta}(t), its short- and long-time limits, the T∞T_\infty decomposition, and the long-time dissipation bound built from the recurrence-time precision asymmetry). The problem asks for the completion of this counting-timing correspondence beyond second moments and at finite times; both parts below are required: (1) establish — or rule out — a deeper and general interrelation between the full probability distribution of Nαβ(t)N_{\alpha\beta}(t) and the distributions of the recurrence times ταβ\tau_{\alpha\beta}, τβα\tau_{\beta\alpha} (for instance by determining which features of the recurrence-time statistics — their asymmetry, higher moments, Kullback-Leibler divergence (the relative entropy between two probability distributions), or the full pair of distributions — are encoded in, or determine, the full counting statistics of the current on generic networks at finite observation time); and (2) derive, directly from the exact channel-resolved expressions for PαβN(t)P^{\mathcal N}_{\alpha\beta}(t), thermodynamic bounds on the channel-specific precision at finite monitoring time tt — TUR-type inequalities involving observable features of the monitored channel (including the precision coefficients of the recurrence times) in addition to the global steady-state entropy-production rate σss\sigma^{ss}. A negative answer to (1) — for example an explicit pair of networks with identical recurrence-time distributions but different full current statistics, or a proof that the second-moment level is the maximal interrelation — is an admissible answer to (1) and closes that part; it does not by itself complete the task, which also requires (2). The source poses the question as whether a deeper interrelation exists, so a refutation of (1) is a legitimate outcome. These two requests are two faces of one objective: the first identifies what event-time information determines the current statistics, and the second converts that channel-resolved information into finite-time dissipation bounds.

References

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