Anomalous transport in external fields: Continuous time random walks and fractional diffusion equations extended

1998 ◽  
Vol 58 (2) ◽  
pp. 1621-1633 ◽  
Author(s):  
Ralf Metzler ◽  
Joseph Klafter ◽  
Igor M. Sokolov
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Anomalous Transport ◽  
Diffusion Equations ◽  
External Fields ◽  
Fractional Diffusion Equations ◽  
Continuous Time Random Walks

scholarly journals Clustered continuous-time random walks: diffusion and relaxation consequences

2012 ◽  
Vol 468 (2142) ◽  
pp. 1615-1628 ◽  
Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Continuous Time ◽  
Waiting Times ◽  
Scaling Limit ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Novel ◽  
Fractional Diffusion Equations ◽  
Continuous Time Random Walks

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

scholarly journals From continuous time random walks to the generalized diffusion equation

2018 ◽  
Vol 21 (1) ◽  
pp. 10-28 ◽  
Author(s):  
Trifce Sandev ◽  
Ralf Metzler ◽  
Aleksei Chechkin
Keyword(s):  
Diffusion Equation ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Probability Distribution Functions ◽  
Special Cases ◽  
Continuous Time Random Walks ◽  
Random Walk Theory

AbstractWe obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.

scholarly journals Understanding anomalous transport in intermittent maps: From continuous-time random walks to fractals

2005 ◽  
Vol 70 (1) ◽  
pp. 63-69 ◽  
Author(s):  
N Korabel ◽  
A. V Chechkin ◽  
R Klages ◽  
I. M Sokolov ◽  
V. Yu Gonchar
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Anomalous Transport ◽  
Continuous Time Random Walks

Continuous time random walk models associated with distributed order diffusion equations

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Sabir Umarov
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Stability Index ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Analytic Method ◽  
Fractional Diffusion Equations ◽  
The Stability ◽  
Distributed Order

AbstractIn this paper continuous time and discrete random walk models approximating diffusion processes associated with time-fractional and spacedistributed order differential equations are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change process is the inverse to a Levy’s stable subordinator with the stability index β ∈ (0, 1). In the paper the convergence of modeled continuous time and discrete random walks to time-changed processes associated with distributed order fractional diffusion equations are proved using an analytic method.

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