Certifying Hidden Dissipation from Observed Current Fluctuations

Technical Summary: Certifying Hidden Dissipation from Observed Current Fluctuations

Problem Statement
In single-molecule experiments involving driven molecular machines (e.g., kinesin motors or enzymes), researchers typically resolve only a subset of the transitions within the underlying state network. While the mechanical steps producing visible displacement are often observable, the chemical substeps driving them remain hidden. The central challenge is to determine the total free energy dissipation (entropy production) of the entire system, including the contributions from these unobserved transitions, without prior knowledge of the transition rates or access to the hidden states. Existing thermodynamic uncertainty relations (TURs) bound dissipation based on chosen currents but fail to specify when partial observation is sufficient to uniquely determine the hidden contribution.

Methodology
The paper analyzes finite-state Markov jump systems in a nonequilibrium steady state (NESS). The approach relies on the statistical properties of empirical currents over long times, specifically their means and fluctuations (covariances).

1. Geometric Framework: The author utilizes the level-2.5 large-deviation rate function for empirical density and flow. By contracting this functional over normalized empirical density fluctuations, they derive a quadratic metric, $M_T$, on the cycle-current subspace. This metric represents the "density-contracted traffic energy," where the local edge metric is the inverse of the dynamical activity (traffic), $1/T_e$.
2. Covariance as a Metric: The asymptotic covariance of empirical edge currents, projected onto the cycle subspace, is shown to be the pseudoinverse of the metric $M_T$.
3. Minimum-Energy Completion: When only a subset of edges $O$ is observed, the observed-current covariance corresponds to a constrained minimization problem. Specifically, the quadratic form $K_O^\top \Sigma_O^+ K_O$ (where $\Sigma_O$ is the observed covariance and $K_O$ the observed mean current) equals the minimum density-contracted traffic energy of any stationary current that reproduces the observed currents on $O$.
4. Cycle Observability: A critical condition is derived: if the observed transitions span the cycle space of the network (i.e., the projection of the cycle basis onto observed edges is injective), the minimum-energy completion is unique and recovers the full network energy.
5. Hybrid Bound: The author combines the quadratic covariance term with the exact nonlinear Schnakenberg entropy production calculated directly from the observed edges. This yields a monotone lower bound on total dissipation that improves as more transitions are observed.

Key Contributions
The paper establishes four primary theoretical advances:

• Cycle-Observability Condition: It identifies a precise condition under which partial observation fixes the hidden contribution to dissipation. Unlike previous TURs that bound dissipation from arbitrary currents, this work proves that if observed edges span the cycle space, the observed-current covariance certifies the full hidden quadratic cost.
• Minimum-Energy Completion Identity: The observed-current covariance is recast as the least-energy stationary completion of hidden currents. This provides a geometric mechanism for how fluctuations encode information about the unobserved network.
• Traffic Metric Identification: The work explicitly identifies the metric within the current covariance object as the density-contracted traffic metric $M_T$. It distinguishes between the raw quadratic traffic cost and the screened cost, introducing a "screening fraction" $\gamma_G(K)$ that quantifies how much traffic cost is absorbed by density fluctuations.
• Operational Monotone Bound: A practical estimator, $B_{O}^{\max}$, is constructed by taking the maximum of the per-edge nonlinear sum and the covariance-based hybrid bound. This estimator is guaranteed to be a lower bound on total entropy production and is monotone non-decreasing as more edges are observed.

Results

• Theoretical Validation: The author demonstrates that when the observed edges span the cycle space, the observed covariance recovers the full density-contracted network energy $K^\top M_T K$. Furthermore, if the physical stationary current is additionally "unscreened" ($\gamma_G(K) = 0$), this recovered energy equals the exact raw quadratic traffic cost $\sum K_e^2/T_e$, even for hidden edges.
• Screening Effects: In regimes where screening is non-zero, the covariance contribution is reduced, but the operational bound remains valid by incorporating the exact nonlinear cost of observed edges.
• Numerical Demonstrations:
  • On a $3 \times 3$ torus network with no screening, the operational bound saturates the true dissipation at full coverage and recovers 93.5% of the total dissipation even when one-third of edges are hidden.
  • On a four-state two-cycle network with varying screening, the bound tracks the predicted theoretical limits, showing that degradation with screening is a structural property of the chain, not a failure of the bound.
• Application to Kinesin: Applying the theory to the Liepelt–Lipowsky six-state kinesin model, the author shows that observing only the mechanical step is insufficient to certify dissipation at stall force (where net current vanishes). However, adding a single chemical transition restores cycle observability, allowing the bound to certify the full cycle energy despite the vanishing net displacement.

Significance
The paper claims that its primary significance lies in transforming partial observation from a source of incomplete information into a solvable "minimum-energy completion" problem. It provides a rigorous certification theorem: if the observed transitions span the network's cycles, the fluctuations of those transitions are sufficient to certify the total hidden dissipation without needing the transition rates or the hidden topology. This bridges the gap between current-fluctuation uncertainty relations and the specific geometric requirements needed to uniquely infer hidden thermodynamic costs. The work emphasizes that the "traffic" (dynamical activity) is the symmetric partner of the current that shapes fluctuations, and this relationship allows for the reconstruction of hidden energetic costs from observable statistics alone.