Cross Spectra Break the Single-Channel Impossibility

2026-04-04

AI Summary

This work addresses the fundamental limitation that single-channel Gaussian time series cannot detect departures from equilibrium in linear systems via time irreversibility. To overcome this, the authors propose a dual-channel observational framework where both channels share a common hidden driving source. By exploiting the geometric structure of the cross-spectrum within the off-diagonal subspace of the spectral matrix, the method transcends the single-channel detection barrier and enables rigorous positive identification of entirely unobservable dissipative signals. The approach integrates cross-spectral analysis, higher-order statistics, stochastic differential equation modeling, and entropy production rate computation. Theoretically, the proposed cross-spectral detectability coefficient is shown to be strictly positive irrespective of the observation timescale—even when timescales coincide—and exhibits a quantitative relationship with the total entropy production rate, thereby providing a solid theoretical foundation for dual-probe experimental designs.

Abstract

Lucente et al. proved that no time-irreversibility measure can detect departure from equilibrium in a scalar Gaussian time series from a linear system. We show that a second observed channel sharing the same hidden driver overcomes this impossibility: the cross-spectral block, structurally inaccessible to any single-channel measure, provides qualitatively new detectability. Under the diagonal null hypothesis, the cross-spectral detectability coefficient $\Scross$ (the leading quartic-order cross contribution) is \emph{exactly} independent of the observed timescales -- a cancellation governed solely by hidden-mode parameters -- and remains strictly positive at exact timescale coalescence, where all single-channel measures vanish. The mechanism is geometric: the cross spectrum occupies the off-diagonal subspace of the spectral matrix, orthogonal to any diagonal null and therefore invisible in any single-channel reduction. For the one-way coupled Ornstein--Uhlenbeck counterpart, the entropy production rate (EPR) satisfies $\EPRtot=α_2λ^2$ exactly; under this coupling geometry, $\Scross>0$ certifies $\EPRtot>0$, linking observable cross-spectral structure to full-system dissipation via $\EPRtot^{\,2}\propto\Scross$. Finite-sample simulations predict a quantitative detection-threshold split testable with dual colloidal probes and multisite climate stations.

Problem

• non-equilibrium detection
• single-channel impossibility
• cross-spectrum
• time irreversibility
• entropy production rate

Innovation

• cross-spectrum
• non-equilibrium detection
• time irreversibility
• entropy production rate
• multi-channel observability