*J. Klafter and I.M. Sokolov*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.003.0003
- Subject:
- Physics, Soft Matter / Biological Physics

Up to now the book has considered the displacement and the return properties of a walker as functions of the number of steps. In physics and chemistry we however are mostly interested in the behavior ...
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Up to now the book has considered the displacement and the return properties of a walker as functions of the number of steps. In physics and chemistry we however are mostly interested in the behavior of the corresponding properties as functions of time. Translating steps into time and back is done by assuming that a walker waits for the next step for a time t distributed according to a known density. Such a process is known as continuous time random walk (CTRW). The CTRW formalism is used to calculate the distribution of displacements, as well as first passage and return times. Special attention is paid to the case when the mean waiting time diverges, as a one often used in applications.Less

Up to now the book has considered the displacement and the return properties of a walker as functions of the number of steps. In physics and chemistry we however are mostly interested in the behavior of the corresponding properties as functions of time. Translating steps into time and back is done by assuming that a walker waits for the next step for a time t distributed according to a known density. Such a process is known as continuous time random walk (CTRW). The CTRW formalism is used to calculate the distribution of displacements, as well as first passage and return times. Special attention is paid to the case when the mean waiting time diverges, as a one often used in applications.

*J. Klafter and I.M. Sokolov*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.003.0008
- Subject:
- Physics, Soft Matter / Biological Physics

The divergent moments of Levy flights are often considered as a handicap for using this model in physics. The remedy is introducing the space‐time coupling, i.e. taking into account that in most ...
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The divergent moments of Levy flights are often considered as a handicap for using this model in physics. The remedy is introducing the space‐time coupling, i.e. taking into account that in most applications a long step takes longer time to perform. The chapter starts from discussing different coupling schemes, and than concentrates on a special model, Levy walk, in which the time cost is proportional to a distance covered. A further generalization is given by a model where such Levy walk is interrupted by rests due to trapping events. Both Levy walks and walks interrupted by rests are relevant for description of experimental situations which are also shortly discussed.Less

The divergent moments of Levy flights are often considered as a handicap for using this model in physics. The remedy is introducing the space‐time coupling, i.e. taking into account that in most applications a long step takes longer time to perform. The chapter starts from discussing different coupling schemes, and than concentrates on a special model, Levy walk, in which the time cost is proportional to a distance covered. A further generalization is given by a model where such Levy walk is interrupted by rests due to trapping events. Both Levy walks and walks interrupted by rests are relevant for description of experimental situations which are also shortly discussed.

*Luis Caffarelli and Luis Silvestre*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0004
- Subject:
- Mathematics, Numerical Analysis

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. ...
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This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels K(t, x, s, y) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.Less

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels *K*(*t*, *x*, *s*, *y*) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.