Department of Physics, Indian Institute of Technology Madras, Chennai, 600036, India.
Phys Rev E. 2017 Sep;96(3-1):032210. doi: 10.1103/PhysRevE.96.032210. Epub 2017 Sep 11.
We study the transport and diffusion properties of passive inertial particles described by a six-dimensional dissipative bailout embedding map. The base map chosen for the study is the three-dimensional incompressible Arnold-Beltrami-Childress (ABC) map chosen as a representation of volume preserving flows. There are two distinct cases: the two-action and the one-action cases, depending on whether two or one of the parameters (A,B,C) exceed 1. The embedded map dynamics is governed by two parameters (α,γ), which quantify the mass density ratio and dissipation, respectively. There are important differences between the aerosol (α<1) and the bubble (α>1) regimes. We have studied the diffusive behavior of the system and constructed the phase diagram in the parameter space by computing the diffusion exponents η. Three classes have been broadly classified-subdiffusive transport (η<1), normal diffusion (η≈1), and superdiffusion (η>1) with η≈2 referred to as the ballistic regime. Correlating the diffusive phase diagram with the phase diagram for dynamical regimes seen earlier, we find that the hyperchaotic bubble regime is largely correlated with normal and superdiffusive behavior. In contrast, in the aerosol regime, ballistic superdiffusion is seen in regions that largely show periodic dynamical behaviors, whereas subdiffusive behavior is seen in both periodic and chaotic regimes. The probability distributions of the diffusion exponents show power-law scaling for both aerosol and bubbles in the superdiffusive regimes. We further study the Poincáre recurrence times statistics of the system. Here, we find that recurrence time distributions show power law regimes due to the existence of partial barriers to transport in the phase space. Moreover, the plot of average particle kinetic energies versus the mass density ratio for the two-action case exhibits a devil's staircase-like structure for higher dissipation values. We explain these results and discuss their implications for realistic systems.
我们研究了由六维耗散避险嵌入映射描述的被动惯性粒子的输运和扩散特性。选择的基映射是三维不可压缩 Arnold-Beltrami-Childress (ABC) 映射，作为体积保持流的表示。有两种截然不同的情况：双作用和单作用情况，这取决于两个参数（A，B，C）中是否有两个或一个超过 1。嵌入映射动力学由两个参数（α，γ）控制，分别量化质量密度比和耗散。气溶胶（α<1）和气泡（α>1）两种情况存在重要差异。我们研究了系统的扩散行为，并通过计算扩散指数 η 构建了参数空间中的相图。根据扩散行为可以将系统分为三类：亚扩散输运（η<1）、正常扩散（η≈1）和超扩散（η>1），其中 η≈2 被称为弹道区。将扩散相图与之前观察到的动力学相图相关联，我们发现超混沌气泡区与正常扩散和超扩散行为密切相关。相比之下，在气溶胶区，在主要表现出周期性动力学行为的区域观察到弹道超扩散，而在周期性和混沌区都观察到亚扩散行为。在超扩散区，气溶胶和气泡的扩散指数概率分布都呈现幂律分布。我们进一步研究了系统的 Poincaré 重游时间统计。在这里，我们发现由于在相空间中存在部分输运障碍，重游时间分布呈现幂律区。此外，对于双作用情况，平均粒子动能与质量密度比的图在较高耗散值下表现出魔鬼阶梯状结构。我们解释了这些结果并讨论了它们对实际系统的意义。
