布尔累积量与更新驱动系统的精确约化描述
Boolean Cumulants and Exact Reduced Descriptions of Renewal-Driven Systems
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中文总结 AI 辅助
本文针对无记忆更新噪声驱动的系统，提出两种精确约化描述：基于布尔累积量的核表示和冻结噪声表示，并揭示闭合误差与平稳密度分别由布尔层级和跳跃分布几何控制。
中文摘要 AI 辅助
对未解析涨落的约化描述通常基于高斯过程，尽管许多实际强迫具有有限相关时间和更新结构。对于无记忆阶跃（Kubo-Anderson）噪声，我们证明约化动力学具有两种精确且互补的描述：一种是由跳跃分布的布尔累积量控制的核表示，另一种是用于平稳概率密度的冻结噪声表示。对于指数分布的等待时间，全时间有序的$G$-累积量与跳跃律的布尔累积量精确重合，从而实现对记忆核的精确重求和。所得核通过布尔生成元$\eta$在预解算子上的取值来表示。在此层级中，当且仅当跳跃分布为对称伯努利分布时，二阶闭合是精确的。对于一般跳跃律，主导闭合误差由普适组合$(b4/b2)\lambda^2$控制。然而，与经典累积量展开不同，布尔层级作用于约化核而非直接作用于平稳测度。一种互补的精确描述由冻结噪声表示提供。对于具有线性乘性相互作用的线性系统（LIMI/CAM）模型，该表示将动力学约化为随机仿射递归，并给出平稳密度和Kesten尾指数的精确表达式。分析表明，速率、矩和闭合误差由布尔层级控制，而平稳密度性质通过冻结动力学由跳跃分布的几何决定。因此，有限相关约化不仅受强迫的方差和相关时间控制，还受更新过程的潜在组合结构控制。对于无记忆更新噪声，该结构是布尔型的。
英文摘要
Reduced descriptions of unresolved fluctuations are commonly based on Gaussian processes, although many realistic forcings have finite correlation times and a renewal structure. For memoryless step (Kubo--Anderson) noise, we show that the reduced dynamics admit two exact and complementary descriptions: a kernel representation governed by the Boolean cumulants of the jump distribution, and a frozen-noise representation adapted to stationary probability densities. For exponentially distributed waiting times, the totally time-ordered $G$-cumulants coincide with the Boolean cumulants of the jump law, and the memory kernel is resummed exactly as the Boolean generator $η$ evaluated on a resolvent operator. Second-order closure is exact if and only if the jump law is symmetric Bernoulli; otherwise, the leading closure error is controlled by $(b_4/b_2)λ^2$. The Boolean hierarchy, however, acts on the kernel, not on the stationary measure. Stationary densities are approximated by replacing the jump law with its $N$-point Gauss quadrature: each surrogate preserves all multi-time correlations up to order $2N-1$ for any waiting-time law, its kernel is a Padé resummation of the Boolean one, and it is itself an exactly solvable renewal problem. For the linear system with linear multiplicative interaction (LIMI/CAM), the frozen-noise representation reduces the dynamics to a random affine recursion and yields exact support boundaries, singularity exponents and Kesten tail indices, confirmed numerically. Rates, moments and closure errors are governed by the Boolean hierarchy, whereas stationary densities are determined by the geometry of the jump distribution. For memoryless renewal noise, the combinatorial structure controlling finite-correlation reductions is Boolean.
发表机构
- CNR–ISMAR（意大利国家研究委员会-海洋科学研究所）
- Dipartimento di Fisica, Università di Pisa（比萨大学物理系）
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