*J. Klafter and I. M. Sokolov*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.001.0001
- Subject:
- Physics, Soft Matter / Biological Physics

The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same ...
More

The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays theory of random walks was proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub‐ and superdiffusive transport processes as well. This book discusses main variants of the random walks and gives the most important mathematical tools for their theoretical description.Less

The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same year, a similar problem was formulated by Albert Einstein in one of his *Annus Mirabilis* works. Even earlier problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays theory of random walks was proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub‐ and superdiffusive transport processes as well. This book discusses main variants of the random walks and gives the most important mathematical tools for their theoretical description.

*Yuri Feldman, Paul Ben Ishai, Alexander Puzenko, and Valerică Raicu*

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780199686513
- eISBN:
- 9780191766398
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199686513.003.0002
- Subject:
- Physics, Soft Matter / Biological Physics

This chapter presents a brief introduction to the modern theory of dielectrics. The different polarization mechanisms in static and time-dependent fields are discussed and the complex dielectric ...
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This chapter presents a brief introduction to the modern theory of dielectrics. The different polarization mechanisms in static and time-dependent fields are discussed and the complex dielectric permittivity and conductivity representations are introduced. The simplest case of the Debye dispersion and different phenomenological non-exponential functions, both in the frequency and time domain, are discussed in detail. The different models of dielectric relaxation are considered, including diffusion and transport in complex dielectrics. The chapter concludes with a discussion of the modern interpretation of the non-Debye relaxation behavior on the basis of anomalous diffusion and percolation theory in fractal lattices.Less

This chapter presents a brief introduction to the modern theory of dielectrics. The different polarization mechanisms in static and time-dependent fields are discussed and the complex dielectric permittivity and conductivity representations are introduced. The simplest case of the Debye dispersion and different phenomenological non-exponential functions, both in the frequency and time domain, are discussed in detail. The different models of dielectric relaxation are considered, including diffusion and transport in complex dielectrics. The chapter concludes with a discussion of the modern interpretation of the non-Debye relaxation behavior on the basis of anomalous diffusion and percolation theory in fractal lattices.

*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/9780199234868.003.0010
- Subject:
- Physics, Soft Matter / Biological Physics

In this chapter another, experimentally widespread, situation is considered. The random walk takes place not on a homogeneous lattice, where each site in principle accessible to the walker, but on a ...
More

In this chapter another, experimentally widespread, situation is considered. The random walk takes place not on a homogeneous lattice, where each site in principle accessible to the walker, but on a percolation structure where some sites are not accessible. Close to the percolation threshold, when the system of accessible sites disintegrates into finite clusters, and the connected way through the whole lattice does not exist anymore, the properties of the corresponding walks are related to the fractal structure of the infinite cluster and are quite unusual. The chapter discusses some basic notions of fractal geometry, the properties of random walks on such structures and their effects on the kinetics of simple reactions in percolation systems. The case when the walker can start at an infinite as well as at a finite cluster is also considered.Less

In this chapter another, experimentally widespread, situation is considered. The random walk takes place not on a homogeneous lattice, where each site in principle accessible to the walker, but on a percolation structure where some sites are not accessible. Close to the percolation threshold, when the system of accessible sites disintegrates into finite clusters, and the connected way through the whole lattice does not exist anymore, the properties of the corresponding walks are related to the fractal structure of the infinite cluster and are quite unusual. The chapter discusses some basic notions of fractal geometry, the properties of random walks on such structures and their effects on the kinetics of simple reactions in percolation systems. The case when the walker can start at an infinite as well as at a finite cluster is also considered.