Directed walks shape a universal square-root law of entropy production rate in nonreciprocal systems

2026-08-25

AI Summary

研究通过定向行走特性表达非互易系统中的熵产生率，并发现其遵循一个普遍的平方根定律，揭示了复杂交互网络如何生成不可逆性。

Abstract

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.

Problem

• Entropy Production Rate
• Nonreciprocal Systems
• Directed Walks
• Interaction Networks

Innovation

• entropy production rate
• directed walks
• antisymmetric matrices
• square-root law
• nonreciprocal systems