嵌入映射中的输运与扩散
Transport and diffusion in the embedding map
摘要：
研究了被动惯性粒子在$2-d$不可压缩流中的输运性质。在这里，粒子动力学是由$2-d$面积保持标准映射的$4-d$耗散嵌入映射表示的，该标准映射模拟不可压缩流。该系统是流体中杂质动力学的模型，其特征在于两个参数，惯性参数$α$和耗散参数$γ$。我们得到了该系统在这些动力学机制下的输运统计特征。这些是，递归时间统计，扩散常数，和跳跃长度的分布。的递归时间分布示出的动力学制度，其中有优先浓度的颗粒在粘性区域的相空间中的幂律尾部，和混合制度的指数衰减。扩散常数显示三种类型的行为-正常的，subdiffusive和superdiffusive，取决于参数制度。构造系统的相图以区分不同类型的扩散行为以及绝对漂移的行为。我们相关的动力学制度，看到在不同的参数值与在这些制度观察到的传输特性的系统，并在瞬态的行为。该系统还显示了危机的存在和不稳定的尺寸可变性在某些参数值。在运输的统计特征中可以看到不稳定尺寸变化的特征。我们讨论了现实系统的影响，我们的结果。
英文摘要：
We study the transport properties of passive inertial particles in a $2-d$ incompressible flows. Here the particle dynamics is represented by the $4-d$ dissipative embedding map of $2-d$ area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter $α$, and the dissipation parameter $γ$. We obtain the statistical characterisers of transport for this system in these dynamical regimes. These are, the recurrence time statistics, the diffusion constant, and the distribution of jump lengths. The recurrence time distribution shows a power law tail in the dynamical regimes where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion constant shows behaviour of three types - normal, subdiffusive and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behaviour, as well as the behaviour of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes, and in the behaviour of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterisers of transport. We discuss the implications of our results for realistic systems.
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