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Directed walks shape a universal square-root law of entropy production rate in nonreciprocal systems
Thiparat Chotibut, Ewa Gudowska-Nowak, Maciej A. Nowak
TL;DR
The paper studies entropy production in linearly stable Ornstein–Uhlenbeck fluctuations and relates its walk expansion to spectral quantities. It identifies a square-root law within the matched-walk and Catalan convergence regime, while distinguishing local Gaussian EPR from full nonlinear EPR.
Problem
The stationary Gaussian fluctuation state is a nonequilibrium steady state when linearized dynamics breaks detailed balance, requiring an account of its local entropy production under stability conditions.
Method
The analysis uses exact walk expansions and, after whitening, preserves eigenvalues and closed-walk traces while imposing matched-walk moment conditions for the square-root prediction.
Results
Under the matched-walk convergence condition g < 1, the spectral calculation contains the same square-root law, whose leading weak-coupling term is J2/2.
Takeaways & Limitations
The walk-counting formulation identifies matched-walk selection and Catalan resummation as the origin of the square-root behavior.
Takeaways & Limitations
The results describe local EPR of Gaussian fluctuations near a fixed point, not the full nonlinear EPR away from that point.
Abstract
from arXiv · showhide
The entropy production rate (EPR) quantifies irreversibility of a nonequilibrium steady state, yet standard formulas obscure how a complex interaction network generates it. For multivariate Ornstein-Uhlenbeck dynamics on such networks, we express the EPR as a quadratic form in antisymmetric matrices measuring the nonreciprocity of aggregate directed walks at every length, and, equivalently, as two weighted-walk quantities: pairs of directed walks sharing both endpoints, and directed closed walks. For diagonalizable interactions, an exact correspondence translates these walk quantities into eigenvalues and biorthogonal eigenvector overlaps. Across dense, sparse, and deep acyclic random interactions satisfying matched-walk conditions, the mean EPR per node universally follows the square-root law $φ_*(g)=1-\sqrt{1-g^2}$, where $g \in [0,1)$ parametrizes the interaction strength. Deep acyclic interaction matrices are nilpotent, with all eigenvalues fixed at zero for every $g$, yet, as their depth increases, their mean EPR per node approaches $φ_*(g)$. Thus, the square-root law arises from directed walk properties, rather than from a shared spectral density or specific network topology.
END MATTER
The end matter connects nonlinear systems to the OU framework, specifies stability and noise-transformation conditions, and compares interaction ensembles through matched coupling strengths. It also relates the square-root EPR law to random neural networks and non-Hermitian spectral results while preserving a local, linearized scope.
- Nonlinear-to-OU translation: Near a stable fixed point, linearized fluctuations of additive nonlinear systems obey OU dynamics with B = −J = I −W.The interaction matrix W is read from the Jacobian, with W = I + J under the paper’s convention.
- Scope and assumptions: The reported EPR describes local Gaussian fluctuations near a fixed point, not the full nonlinear EPR away from that point.The convergent walk expansion additionally requires ρ(W) = ρ(I + J) < 1.
- Noise generality: Any constant positive-definite diffusion matrix can be whitened without changing stationary EPR, but the square-root prediction still requires matched-walk moments after whitening.Whitening preserves eigenvalues, spectral radius, and closed-walk traces, but not generally Frobenius norms or walk-pair traces.
- Ensemble parametrization: The compared ensembles share asymptotic mean-squared interaction strength per node g^2 in the large-N or deep-network limits.The ensemble parametrization includes recurrent networks, out-trees, and L-layer feedforward networks, with the limiting square-root resummation recorded alongside the matched-walk analysis.
- Neural-network connection: For random neural networks, the linearized OU result agrees with the nonlinear theory at weak coupling, while the complete nonlinear EPR differs from the OU curve at finite-noise critical coupling.The weak-coupling leading term is J^2/2, and the OU curve approaches 1 as J → 1−.
- Spectral connection: A non-Hermitian spectral calculation contains the same square-root law after time normalization, while walk counting identifies matched-walk selection and Catalan resummation as its origin.The spectral correspondence applies to the Ginibre-related linear neural-network setting under its stated stability and normalization conditions.