Ordered Absorption Reveals Green-Kubo Memory Hidden from Survival Curves

Technical Summary: Ordered Absorption Reveals Green-Kubo Memory Hidden from Survival Curves

Problem Statement
Many physical experiments, particularly in single-molecule biophysics and transport phenomena, register only "loss" or "survival" events (e.g., escape from a trap or depletion of a state) while the underlying continuous trajectory remains inaccessible. While absorbing boundaries are known to connect to escape and transport, and Green-Kubo relations rely on time correlations to describe response, a fundamental limitation exists: static survival curves generated by a fixed, state-dependent absorber are identical for a process and its time reverse. Consequently, stationary irreversibility and dynamical memory are fundamentally absent from standard survival measurements, regardless of observation time or loading strength. The paper asks what minimal temporal control is required to restore this hidden information.

Methodology
The authors propose a theoretical framework and retrospective validation using "ordered absorption."

1. Theoretical Framework:

   • The model considers a stationary process $X_t$ subjected to calibrated pulses with profiles $a$ and $b$ at times $0$ and $n$, respectively. These pulses remove a realization with probability proportional to the state, without altering the dynamics of survivors.
   • The authors prove that while a single survival curve is time-reversal invariant, the joint survival probability of two ordered pulses, $S_{ab}(n)$, contains the necessary information.
   • They derive exact identities showing that the difference between forward and reversed pulse sequences, $S_{ab}(n) - S_{ba}(n)$, yields the lagged covariance $\text{Cov}(a_0, b_n)$ exactly, without requiring Markov assumptions, mixing conditions, weak-pulse approximations, or quasistationary limits.
   • By using spanning profiles (a set of linearly independent profiles), the authors demonstrate that the finite-state transition matrix of a Markov model can be reconstructed. Furthermore, they establish a bound on path irreversibility using Bernoulli relative entropy, $d_B(S_{ab} \| S_{ba})$, which certifies time's arrow without full reconstruction.

2. Experimental Validation (Retrospective):

   • The authors applied this framework to 20 independent optical-trap records (SwitchingTrap subset) containing bead position and force data at 100 kHz.
   • They computationally emulated the stochastic removal pulses using the first 20% of each record to define robust logistic profiles, then applied these "virtual pulses" to the held-out 80% of the data.
   • They compared the "survival-order" estimator (derived from virtual survival probabilities) against the direct covariance calculated from the full trajectory.
   • A reversible two-state single-molecule model (HP3) was used as a null benchmark to test reconstruction accuracy and false-positive rates.

Key Contributions

• Proof of Invariance: Demonstrated that any stationary survival curve generated by a fixed absorber is invariant under time reversal, proving that static survival measurements cannot reveal dynamical memory.
• Exact Recovery via Two Pulses: Proved that two ordered pulses with known profiles allow for the exact recovery of finite-amplitude lagged covariances and the time-antisymmetric component of the dynamics.
• Reconstruction and Certification: Showed that spanning profiles can reconstruct finite-state Markov dynamics and that the resulting Bernoulli probabilities provide a rigorous bound on path irreversibility via relative entropy.
• Green-Kubo Recovery: Established that these measurements can recover the Green-Kubo spectral kernel, with the order contrast supplying the antisymmetric (imaginary) cross-component.

Results

• Experimental Data: In the optical-trap records, the normalized experimental time-antisymmetric correlation was robustly detected at 10 ms lag for various trap separations (18, 70, and 280 nm). The "survival-order" estimates derived from virtual pulses agreed with direct covariance calculations to machine precision after applying finite-record marginal corrections.
• Information Contraction: The study confirmed that the Bernoulli survival channel obeys $d_B(S_{ab} \| S_{ba}) \leq D(P_{ij} \| P_{ji})$, demonstrating information contraction consistent with coarse observation principles.
• Finite-Count Sensitivity: Simulations indicated that detecting order asymmetry requires significant trial counts (e.g., detection probability rises from ~8% to ~96% as trials per probability increase from $10^5$ to $10^6$), highlighting the cost of coarse readout.
• Null Benchmark: For the reversible HP3 two-state model, the survival-only reconstruction showed decreasing Frobenius error with increasing trial counts ($M$), and the reversible order test maintained a conservative false-positive rate near $10^{-3}$.

Significance and Claims
The paper claims that while static survival curves are "exactly blind" to stationary time reversal, the introduction of two calibrated, ordered removals is sufficient to reveal lagged memory and projected currents. This method restores identifiability to systems where only loss events are observable, effectively recovering Green-Kubo memory hidden from standard survival analysis.

The authors emphasize that their work is a retrospective analysis using experimental trajectories to computationally emulate the proposed removals; they explicitly state that "direct physical pulse validation remains open." The significance lies in defining the minimal control (two ordered pulses) required to break the symmetry of survival measurements and recover dynamical information, providing a theoretical benchmark for future physical implementations in single-molecule and transport experiments.