Transport properties of the continuous-time random walk with a long-tailed waiting-time density

1989 ◽  
Vol 57 (1-2) ◽  
pp. 301-317 ◽  
Author(s):  
Haim Weissman ◽  
George H. Weiss ◽  
Shlomo Havlin
Keyword(s):  
Random Walk ◽  
Transport Properties ◽  
Waiting Time ◽  
Continuous Time ◽  
Continuous Time Random Walk ◽  
Time Density

scholarly journals Comb Model: Non-Markovian versus Markovian

2019 ◽  
Vol 3 (4) ◽  
pp. 54 ◽  
Author(s):  
Alexander Iomin ◽  
Vicenç Méndez ◽  
Werner Horsthemke
Keyword(s):  
Wave Function ◽  
Random Walk ◽  
Transport Properties ◽  
Power Law ◽  
Continuous Time ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Continuous Time Random Walk ◽  
Slow Diffusion ◽  
Branched Structures

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study two generalizations of comb models and present a generic method to obtain their transport properties. The first is a continuous time random walk on a many dimensional m + n comb, where m and n are the dimensions of the backbone and branches, respectively. We observe subdiffusion, ultra-slow diffusion and random localization as a function of n. The second deals with a quantum particle in the 1 + 1 comb. It turns out that the comb geometry leads to a power-law relaxation, described by a wave function in the framework of the Schrödinger equation.

scholarly journals A Directed Continuous Time Random Walk Model with Jump Length Depending on Waiting Time

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Long Shi ◽  
Zuguo Yu ◽  
Zhi Mao ◽  
Aiguo Xiao
Keyword(s):  
Random Walk ◽  
Probability Density Function ◽  
Laplace Transform ◽  
Probability Density ◽  
Density Function ◽  
Waiting Time ◽  
Continuous Time ◽  
Dimensional Space ◽  
Continuous Time Random Walk ◽  
Random Walk Model

In continuum one-dimensional space, a coupled directed continuous time random walk model is proposed, where the random walker jumps toward one direction and the waiting time between jumps affects the subsequent jump. In the proposed model, the Laplace-Laplace transform of the probability density functionP(x,t)of finding the walker at positionxat timetis completely determined by the Laplace transform of the probability density functionφ(t)of the waiting time. In terms of the probability density function of the waiting time in the Laplace domain, the limit distribution of the random process and the corresponding evolving equations are derived.

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

Soft Matter ◽  
2021 ◽  
Author(s):  
Jian Liu ◽  
Caiyun Zhang ◽  
Jing-Dong Bao ◽  
Xiaosong Chen
Keyword(s):  
Random Walk ◽  
Velocity Field ◽  
Waiting Time ◽  
Continuous Time ◽  
Fluid Velocity ◽  
Time Correlation ◽  
Space Time ◽  
Continuous Time Random Walk ◽  
Weak Asymptotics

Within the framework of space-time correlated continuous-time random walk model, anomalous diffusion of particle 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. 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. Numerical results indicate that the multi-fractional diffusion leads to a crossover phenomenon in-between the process at 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 make the diffusion to be characterized by diffusion coefficient instead of diffusion exponent, which is distinctly different from the former definition. Especially, for the waiting time displaying weak asymptotic property, if the anomalous part is suppressed by the normal part, a second crossover phenomenon appears at intermediate timescale, followed by a steady normal diffusion, which implies that the anomalies underlying the process are smoothed out at large timescale. 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 for the relevant experimental research.

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