有向游走塑造非互易系统中熵产生率的普适平方根定律
Directed walks shape a universal square-root law of entropy production rate in nonreciprocal systems
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中文总结 AI 辅助
该研究揭示非互易系统中平均节点熵产生率遵循普适平方根定律，其源于有向游走特性而非网络拓扑，通过多元Ornstein-Uhlenbeck动力学的游走量与特征值对应关系推导得出。
中文摘要 AI 辅助
熵产生率（EPR）量化非平衡稳态的不可逆性，但标准公式模糊了复杂相互作用网络产生熵产生率的机制。对于此类网络上的多元Ornstein-Uhlenbeck动力学，我们将熵产生率表示为反对称矩阵的二次型，这些反对称矩阵测量不同长度下聚合有向游走的非互易性；等价地，也可表示为两种带权游走量：共享两端点的有向游走对，以及有向闭合游走。对于可对角化的相互作用，精确对应关系将这些游走量转化为特征值与双正交特征向量重叠。在满足匹配游走条件的稠密、稀疏及深度无环随机相互作用中，平均每个节点的熵产生率普遍遵循平方根定律φ₊(g)=1−√(1−g²)，其中g∈[0,1)为相互作用强度参数。深度无环相互作用矩阵为幂零矩阵，对任意g其所有特征值均固定为0，但随着深度增加，平均每个节点的熵产生率趋近于φ₊(g)。因此，平方根定律源于有向游走特性，而非共享谱密度或特定网络拓扑。
英文摘要
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.
