Book contents
- Frontmatter
- Contents
- Preface
- Part one Basic concepts
- 1 Fractals
- 2 Percolation
- 3 Random walks and diffusion
- 4 Beyond random walks
- Part two Anomalous diffusion
- Part three Diffusion-limited reactions
- Part four Diffusion-limited coalescence: an exactly solvable model
- Appendix A The fractal dimension
- Appendix B The number of distinct sites visited by random walks
- Appendix C Exact enumeration
- Appendix D Long-range correlations
- References
- Index
3 - Random walks and diffusion
Published online by Cambridge University Press: 19 January 2010
- Frontmatter
- Contents
- Preface
- Part one Basic concepts
- 1 Fractals
- 2 Percolation
- 3 Random walks and diffusion
- 4 Beyond random walks
- Part two Anomalous diffusion
- Part three Diffusion-limited reactions
- Part four Diffusion-limited coalescence: an exactly solvable model
- Appendix A The fractal dimension
- Appendix B The number of distinct sites visited by random walks
- Appendix C Exact enumeration
- Appendix D Long-range correlations
- References
- Index
Summary
Random walks model a host of phenomena and find applications in virtually all sciences. With only minor adjustments they may represent the thermal motion of electrons in a metal, or the migration of holes in a semiconductor. The continuum limit of the random walk model is known as “diffusion”. It may describe Brownian motion of a particle immersed in a fluid, as well as heat propagation, the spreading of a drop of dye in a glass of still water, bacterial motion and other types of biological migration, or the spreading of diseases in dense populations. Random-walk theory is useful in sciences as diverse as thermodynamics, crystallography, astronomy, biology, and even economics, in which it models fluctuations in the stock market.
The simple random walk
A random walk is a stochastic process defined on the points of a lattice. Usually, the time variable is considered discrete. At each time unit the “walker” steps from its present position to one of the other sites of the lattice according to a prescribed random rule. This rule is independent of the history of the walk, and so the process is Markovian.
In the simplest version of a random walk, the walk is performed in a hypercubic d-dimensional lattice of unit lattice spacing. At each time step the walker hops to one of its nearest-neighbor sites, with equal probabilities.
- Type
- Chapter
- Information
- Publisher: Cambridge University PressPrint publication year: 2000
- 1
- Cited by