短时间编织指数
Finite-Time Braiding Exponents
摘要：
拓扑熵是流动中混合速率的常用度量。它可以通过分区方法计算，或者通过估计物质线或其他物质元素的增长率来计算。这需要对速度场有详细的了解，而这并不总是可用的，例如当我们只知道一些粒子轨迹时（例如海洋漂浮物数据）。我们提出了另一种近似拓扑熵，适用于二维流，它只使用有限数量的轨迹作为输入数据。为了表示这些稀疏的数据集，我们使用辫子，代数对象，记录如何股，即，轨迹，相对于投影轴交换位置。由流动平流输送的材料曲线表示为简化的回路坐标。当股线交换位置时，编织物在有限时间间隔内变形成环的指数速率是时间编织指数（FTBE），并且用作二维流的拓扑熵的代理。我们证明了FTBE是强大的数值时间步长，编织表示的细节，并选择混合区域内的初始条件的值。我们还探讨了如何密切的FTBE近似拓扑熵取决于所使用的轨迹的数量和长度。
英文摘要：
Topological entropy is a common measure of the rate of mixing in a flow. It can be computed by partition methods, or by estimating the growth rate of material lines or other material elements. This requires detailed knowledge of the velocity field, which is not always available, such as when we only know a few particle trajectories (ocean float data, for example). We propose an alternative approximation to topological entropy,applicable to two-dimensional flows, which uses only a finite number of trajectories as input data. To represent these sparse data sets, we use braids, algebraic objects that record how strands, i.e., trajectories, exchange positions with respect to a projection axis. Material curves advected by the flow are represented as simplified loop coordinates. The exponential rate at which a braid deforms loops over a finite time interval as the strands exchange places is the Finite-Time Braiding Exponent (FTBE) and serves as a proxy for topological entropy of the two-dimensional flow. We demonstrate that FTBEs are robust with respect to the value of numerical time step,details of braid representation, and choice of initial conditions inside the mixing region. We also explore how closely the FTBEs approximate topological entropy depending on the number and length of trajectories used.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
