Explaining Continuous Time Random Walks: What Is It and How Is It Used?

[yaboidjaf]

Hi, I'm trying to read into CTRW, but I'm finding the information online a little difficult to take in. From what I've read the process differs from normal random walks in that jumps take place after some waiting time $$\tau$$, which can be from 0<$$\tau$$<$$\infty$$. Would I also be right in saying that the jump distance is continuous as well?

After that I come to the Montroll-Weiss equation, Fourier and Laplace transforms and work with probabilities about the positioning after a certain number of steps or time. I'm really looking for something that explains the meaning of all the maths, what the probabilities and transforms show about a particle undergoing this process. Maybe a practical application or a simpler explanation even? Any suggested sources?

Any help would be much appreciated. Thanks.

 

[D H]

Hmm. My initial thought when I read your post was "Yikes! Yet another name for a Gauss-Markov process"! But you are talking about something else.

I found this paper; if you've already come across it, never mind!

Meerschaert et al, Governing equations and solutions of anomalous random walk limits, PRevE, 66:060102
http://inside.mines.edu/~dbenson/current/couplePRE.pdf

 

[Andy Resnick]

AFAICT, the main motivation for developing this line of study is to model "anomalous diffusive transport". That is, diffusive processes where the mean squared displacement <x^2> != t, but is instead a power law t^a.

This situation occurs for diffusion in disordered media:
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVP-46SXPMN-7F&_user=7774119&_coverDate=11%2F30%2F1990&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000062847&_version=1&_urlVersion=0&_userid=7774119&md5=7f2d1c134653937175696ea89d5db97d&searchtype=a

cell migration:
http://www.mpipks-dresden.mpg.de/~rklages/publ/publ.html[/URL]

searching patterns of wandering animals:
[url]http://www.nature.com/nature/journal/v449/n7165/abs/nature06199.html[/url]

and intracellular transport:
[url]http://biophysics.physics.brown.edu/BPJC/JC%20pdf%20paper%20files/BPJC%20Fall%202005/CaspiPRL2000.pdf[/url]

 

[rkla]

Hi,
I found this discussion, because Andy linked to my publications (thanks!).
Perhaps Section 4 of a review is helpful that I wrote last year. It got published as a book chapter, http://arxiv.org/abs/0804.3068" .
It contains a number of further introductory references.
Good luck,
R.

 

[pmacias]

Hello,

Thank you for your interest in continuous time random walks (CTRW). CTRW is a mathematical model used to describe the movement of particles or agents in a non-Markovian system, where the jump lengths and waiting times between jumps are not independent. The concept of CTRW was first introduced by Montroll and Weiss in 1965, and it has since been applied in various fields such as physics, biology, and finance.

In CTRW, the jump lengths and waiting times are considered to be random variables, which can be described by probability distributions. The waiting time \tau is the time between consecutive jumps, and it can take any value between 0 and infinity. This means that the jumps in CTRW can occur at any time, not just at discrete time intervals like in a regular random walk. The jump distance can also be continuous, meaning that the particle can move any distance, not just discrete steps.

The Montroll-Weiss equation is a mathematical equation that describes the probability distribution of the particle's position after a certain number of jumps or time steps. This equation is derived using Fourier and Laplace transforms, which are mathematical tools used to analyze and manipulate probability distributions. The resulting probability distribution gives information about the particle's position and how it changes over time in the CTRW process.

As for practical applications, CTRW has been used to model various phenomena such as diffusion in disordered media, transport in porous materials, and animal movement patterns. It has also been used in finance to model stock price movements and in biology to study the movement of cells and organisms.

If you would like a simpler explanation of CTRW, I would suggest looking into introductory materials on random walks and Markov processes. Understanding these concepts will provide a foundation for understanding CTRW. Additionally, there are many research papers and books available that delve deeper into the mathematics and applications of CTRW.

I hope this helps to clarify some of your questions about CTRW. If you have any further questions, please don't hesitate to ask. Best of luck in your studies!