可压缩Baker映射的随机游动等价性及其信息维的Kaplan-Yorke逼近
Random Walk Equivalence to the Compressible Baker Map and the Kaplan-Yorke Approximation to Its Information Dimension
摘要：
简单的时间可逆系统可以产生满足热力学第二定律的{ \it不可逆}流动。映射和等价的随机游走也可以做到这一点。研究了一对时间可逆的Baker映射$N2$和$N3$，它们产生了耗散的{ \it分形}相空间结构.相空间体积的稳定减小对应于与熵产生相关的耗散。就像20世纪80年代和90年代开发的三个光滑可逆耗散单体相空间流一样，我们的地图生成分形分布，但在二维而不是三维中，简化了可视化和分析。量化相体积损失的连续性方程激发了对分形“简化的”信息维数“的研究，Kaplan和Yorke用二维映射的两个李雅普诺夫指数来近似信息维数。这里研究的地图生成分形（分数维）分布在其相空间。通过映射点的均匀密集网格，可以通过“按面积”映射来确定分形维数。从$N2$ Baker地图的均匀网格区域映射开始，提供了1.78969的信息维度。或者，从任意点开始，多达万亿次的迭代给出了更小的“逐点”维数$1.741_5$。这些精确确定的估计值都不符合卡普兰-约克猜想值1.7337。在研究这三种不同的信息维方法的过程中，我们开发了两种映射的随机游走等价物，这大大简化了分析。我们发现，对于较旧的$ N2$ Baker映射，这三种方法都不一致！我们后来发现，对于较新的$N3$ Baker映射，信息维的三种方法，区域方式，点方式和卡普兰-约克，是一致的。
英文摘要：
Simple time-reversible systems can generate {\it irreversible} flows satisfying the Second Law of Thermodynamics. Maps, and equivalent random walks, can also do this. We study a pair of time-reversible Baker Maps, $N2$ and $N3$, which generate dissipative{\it fractal} phase-space structures. Steadily decreasing phase-space volumes correspond to the dissipation associated with entropy production. Like three smooth reversible dissipative one-body phase-space flows developed in the 1980s and 1990s our maps generate fractal distributions, but in two dimensions rather than three, simplifying visualization and analyses. The continuity equation, which quantifies phase-volume loss, motivates study of the fractals' reduced ``information dimensions'', which were approximated by Kaplan and Yorke in terms of two-dimensional maps' two Lyapunov exponents. The maps studied here generate fractal (fractional dimensional) distributions in their phase spaces. By mapping uniformly dense grids of points, fractal dimensions can be determined by ``area-wise'' mappings. Beginning with a uniform grid area-wise mapping of the $N2$ Baker Map provides an information dimension of 1.78969. Alternatively, as many as a trillion iterations, starting from an arbitrary point, gives a smaller ``point-wise'' dimensionality, $1.741_5$. Neither of these precisely determined estimates matches the Kaplan-Yorke conjecture value, 1.7337. In the course of studying these three different approaches to information dimension we developed random walk equivalents to both mappings, which greatly simplifies analyses. We found that for the older $ N2$ Baker map the three approaches all disagree with one another! We later discovered that for the newer $N3$ Baker mapping the three approaches to information dimension, area-wise, point-wise and Kaplan-Yorke, agree.
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