We consider the dynamics of a separable Continuous Time Random Walk (CTRW) when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding universe. The considered CTRW can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Levy flight. We first consider the case of zero velocity field. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Levy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: In one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.

中文翻译：

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速度场中的连续时间随机游走：域增长、伽利略不变平流扩散和粒子混合动力学的作用

当随机游走器被均匀增长域中的速度场偏置时，我们考虑可分离连续时间随机游走 (CTRW) 的动力学。这些领域的具体例子包括不断增长的生物细胞或脂质囊泡、生物膜和组织，以及宏观系统，如雨季膨胀的含水层或膨胀的宇宙。考虑的 CTRW 可以是亚扩散、正常扩散或超扩散，包括 Levy 飞行的特殊情况。我们首先考虑零速度场的情况。在亚扩散情况下，我们揭示了粒子概率密度函数峰态的有趣时间依赖性。特别是，对于合适的参数选择，我们发现短时间肥尾的传播器可能会交叉到类高斯传播器。我们随后结合了速度场的影响并推导出了一个双分数扩散平流方程，该方程编码了粒子分布的时间演变。我们应用这个方程来研究两个扩散脉冲的混合动力学，在相反方向的速度场的作用下，它们的峰值相互移动。峰的这种确定性运动，连同每个脉冲的扩散扩展，往往会增加粒子混合，从而抵消域增长引起的峰分离。由于这种竞争，出现了不同的混合制度。在 Levy 飞行的情况下，除了非混合状态外，在长时间限制内有两种不同的混合状态，这取决于确切的参数选择：在这些状态之一中，混合主要由扩散传播驱动，而在另一种情况下，混合由作用于每个脉冲的速度场控制。讨论了实际系统中遭遇控制反应的可能影响。