Correlated continuous-time random walk in the velocity field: the role of velocity and weak asymptotics

• Published in: Soft matter 2021-11, Vol.17 (42), p.9786-9798
• Main Authors: Liu, Jian, Zhang, Caiyun, Bao, Jing-Dong, Chen, Xiaosong
• Format: Article
• Language: English
• Subjects: Anomalies; Asymptotic properties; Bias; Correlation; Diffusion; Diffusion coefficient; Experimental research; Random walk; Spacetime; Time; Velocity; Velocity distribution
• ISSN: 1744-683X; 1744-6848
• DOI: 10.1039/d1sm00995h

Within the framework of a space-time correlated continuous-time random walk model, anomalous diffusion of particles moving in the velocity field is studied in this paper. The weak asymptotic form ω ( t ) ∼ t −(1+ α ) , 1 < α < 2 for large t , is considered to be the waiting time distribution. The analytical results reveal that the diffusion in the velocity field, i.e. , the mean squared displacement, can display a multi-fractional form caused by dispersive bias and space-time correlation. The numerical results indicate that the multi-fractional diffusion leads to a crossover phenomenon in-between the process at an intermediate timescale, followed by a steady state which is always determined by the largest diffusion exponent term. In addition, the role of velocity and weak asymptotics is discussed. The extremely small fluid velocity can characterize the diffusion by a diffusion coefficient instead of diffusion exponent, which is distinctly different from the former definition. In particular, for the waiting time displaying a weak asymptotic property, if the anomalous part is suppressed by the normal part, a second crossover phenomenon appears at an intermediate timescale, followed by a steady normal diffusion, which implies that the anomalies underlying the process are smoothed out at large timescales. Moreover, we discuss that the consideration of bias and correlation could help to avoid a possible not readily noticeable mistake in studying the topic concerned in this paper, which may be helpful in the relevant experimental research. Within the framework of a space-time correlated continuous-time random walk model, anomalous diffusion of particle moving in the velocity field is studied. The dispersive bias and space-time correlation lead to a crossover phenomenon in-between the diffusion. While, the weak asymptotics of the waiting time can yield the second unexpected one.