标题： 混沌系统中动力学可观测量分布的显著普适性 显示英文标题

标题： Remarkable universalities in distributions of dynamical observables in chaotic systems

摘要： 对混沌系统的研究，其中罕见事件起着关键作用，对于理解由于初始条件的敏感性而产生的复杂动力学至关重要。 最近，通常应用于随机过程的大偏差理论工具已被用于混沌系统的研究。 在这里，我们研究沿混沌轨迹$A = \sum_{n=1}^N g(\textbf{x}_n)$定义的动力学可观测量$\{\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_N\}$。 对于大多数$g(\textbf{x})$的选择，$A$满足中心极限定理：在序列长度较大时$N \gg 1$，$A$的典型波动遵循方差随$N$线性增长的高斯分布。 大偏差的$A$通常由大偏差原理来描述，即$P(A) \sim e^{- N I(A/N)}$，其中$I(a)$是速率函数。 我们发现某些动力学可观测量表现出显著的统计普适性：即使由不同的函数$g_1(\textbf{x})$和$g_2(\textbf{x})$构造，不同的可观测量也由相同的速率函数描述。 我们通过证明$g_1(\textbf{x})-g_2(\textbf{x})$属于我们称之为“派生”的函数类，为这种惊人的普适性提供了物理解释。 此外，我们证明如果$g(\textbf{x})$本身是“导出”的，那么在大$N$极限下，$A$的分布变得与$N$无关，并且通常是非高斯的（尽管它是镜像对称的）。我们演示了逻辑映射的有限时间李雅普诺夫指数（FTLE）具有这种导出形式，从而为一些现有结果提供了一个简单的解释。 显示英文摘要

摘要： The study of chaotic systems, where rare events play a pivotal role, is essential for understanding complex dynamics due to their sensitivity to initial conditions. Recently, tools from large deviation theory, typically applied in the context of stochastic processes, have been used in the study of chaotic systems. Here, we study dynamical observables, $A = \sum_{n=1}^N g(\textbf{x}_n)$, defined along a chaotic trajectory $\{\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_N\}$. For most choices of $g(\textbf{x})$, $A$ satisfies a central limit theorem: At large sequence size $N \gg 1$, typical fluctuations of $A$ follow a Gaussian distribution with a variance that scales linearly with $N$. Large deviations of $A$ are usually described by the large deviation principle, that is, $P(A) \sim e^{- N I(A/N)}$, where $I(a)$ is the rate function. We find that certain dynamical observables exhibit a remarkable statistical universality: even when constructed with distinct functions $g_1(\textbf{x})$ and $g_2(\textbf{x})$, different observables are described by the same rate function. We provide a physical interpretation for this striking universality by showing that $g_1(\textbf{x})-g_2(\textbf{x})$ belongs to a class of functions that we call ``derived''. Furthermore, we show that if $g(\textbf{x})$ itself is ``derived'', then the distribution of $A$ becomes independent of $N$ in the large-$N$ limit, and is generally non-Gaussian (although it is mirror-symmetric). We demonstrate that the finite-time Lyapunov exponent (FTLE) for the logistic map is of this derived form, thus providing a simple explanation for some existing results.