Heseltine James, Kim Eun-Jin
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK.
Entropy (Basel). 2019 Aug 8;21(8):775. doi: 10.3390/e21080775.
It is often the case when studying complex dynamical systems that a statistical formulation can provide the greatest insight into the underlying dynamics. When discussing the behavior of such a system which is evolving in time, it is useful to have the notion of a metric between two given states. A popular measure of information change in a system under perturbation has been the relative entropy of the states, as this notion allows us to quantify the difference between states of a system at different times. In this paper, we investigate the relaxation problem given by a single and coupled Ornstein-Uhlenbeck (O-U) process and compare the information length with entropy-based metrics (relative entropy, Jensen divergence) as well as others. By measuring the total information length in the long time limit, we show that it is only the information length that preserves the linear geometry of the O-U process. In the coupled O-U process, the information length is shown to be capable of detecting changes in both components of the system even when other metrics would detect almost nothing in one of the components. We show in detail that the information length is sensitive to the evolution of subsystems.
在研究复杂动力系统时，通常情况下统计公式能够最深入地洞察潜在动力学。在讨论这样一个随时间演化的系统的行为时，拥有两个给定状态之间的度量概念是很有用的。系统在扰动下信息变化的一种常用度量是状态的相对熵，因为这个概念使我们能够量化系统在不同时刻状态之间的差异。在本文中，我们研究由单个和耦合的奥恩斯坦 - 乌伦贝克（O - U）过程给出的弛豫问题，并将信息长度与基于熵的度量（相对熵、詹森散度）以及其他度量进行比较。通过在长时间极限下测量总信息长度，我们表明只有信息长度保留了O - U过程的线性几何结构。在耦合的O - U过程中，即使其他度量在其中一个分量中几乎检测不到任何变化，信息长度也能够检测到系统两个分量中的变化。我们详细表明信息长度对子系统的演化很敏感。
