Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx) Universidad de Extremadura, E-06071 Badajoz, Spain.
Departamento de Física Aplicada and Instituto de Computación Científica Avanzada (ICCAEx) Centro Universitario de Mérida Universidad de Extremadura, E-06800 Mérida, Spain.
Phys Rev E. 2019 Jul;100(1-1):012142. doi: 10.1103/PhysRevE.100.012142.
We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing or contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive continuous-time random walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the aging of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing or contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.
我们研究了在均匀生长或收缩域上的调和势（Ornstein-Uhlenbeck 过程）中的正常扩散和亚扩散过程。我们的出发点是最近推导出的分数阶福克-普朗克方程，它涵盖了布朗扩散和亚扩散连续时间随机游走（CTRW）的情况。我们发现随机游走性质对域增长率细节非常敏感，这导致了具有极其不同行为的各种状态。这种丰富的现象学的起源是，由于域生长引起的确定性漂移，即使在等待跳跃时，游走者仍然在移动。因此，与亚扩散 CTRW 的老化相关的越来越长的等待时间意味着，在两次连续跳跃之间的时间间隔内，游走者可能会走过比正常扩散情况下长得多的距离。这导致了看似违反直觉的效应。例如，在静态域上，当存在调和势时，布朗扩散和亚扩散 CTRW 都会导致具有有限宽度的稳定粒子分布，从而表明扩散粒子受到限制。然而，对于足够快速的生长或收缩域，这种定性行为会失效，并且会发现布朗情况和亚扩散情况之间的差异。对于布朗粒子，需要足够快速的指数域增长才能打破调和力引起的限制；相比之下，对于亚扩散粒子，对于足够快速的幂律域增长，这种破坏可能已经发生。我们对这两种扩散类型的分析和数值结果都通过随机游走模拟得到了完全证实。
