Book Section
Stochastic Models
- Title
- Stochastic Models
- Creator(s)
- Ferrari, P. A.
- Editor(s)
- Smelser, Neil J.
- Baltes, Paul B.
- Date
- 2001
- Is Part Of
- International Encyclopedia of the Social & Behavioral Sciences
- Pages
- 15121-15126
- Language
- eng
- Publisher
- Pergamon
- Place Published
- Oxford
- ISBN
- 978-0-08-043076-8
- Abstract
- Paradigmatic examples of stochastic processes are coin-tossing and the sequences of uniform random numbers provided by computer routines. A large number of independent random experiments show nontrivial collective phenomena such as the deterministic behavior of averages, known as the law of large numbers and qualitative changes as a consequence of small quantitative parameter changes known as phase transitions. The behavior of the number of individuals of a population may be described by birth-and-death processes, for which at each unit of time a new individual is born or a present individual dies, and by branching processes, for which each new individual generates a family that grows and dies independently of the other families. These examples are particular cases of Markov chains roughly described by the fact that the probabilistic law of the next experiment depends only on the result of the current one. The main issue for these chains is the study of their long time behavior. Interacting particle systems refer to the time evolution of families of processes for which the updating of each member of the family depends on the current values of the other members. The voter model and the exclusion process are discussed. Hydrodynamics deals with the study of particle systems in large space regions at long times relating the stochastic systems with deterministic partial differential equations.
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