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3 - Interacting stochastic particle systems

Published online by Cambridge University Press:  05 June 2014

By
Stefan Grosskinsky
Edited by
Robin Ball ,
Vassili Kolokoltsov  and
Robert S. MacKay
Stefan Grosskinsky
Affiliation:
University of Warwick
Robin Ball
Affiliation:
University of Warwick
Vassili Kolokoltsov
Affiliation:
University of Warwick
Robert S. MacKay
Affiliation:
University of Warwick
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Summary

Abstract

Interacting particle systems (IPS) are probabilistic mathematical models of complex phenomena involving a large number of interrelated components. There are numerous examples within all areas of natural and social sciences, such as traffic flow on motorways or communication networks, opinion dynamics, spread of epidemics or fires, genetic evolution, reaction diffusion systems, crystal surface growth, financial markets, etc. The central question is to understand and predict emergent behaviour on macroscopic scales, as a result of the microscopic dynamics and interactions of individual components. Qualitative changes in this behaviour depending on the system parameters are known as collective phenomena or phase transitions and are of particular interest.

In IPS the components are modelled as particles confined to a lattice or some discrete geometry. But applications are not limited to systems endowed with such a geometry, since continuous degrees of freedom can often be discretized without changing the main features. So depending on the specific case, the particles can represent cars on a motorway, molecules in ionic channels, or prices of asset orders in financial markets (see Chapter 6), to name just a few examples. In principle, such systems often evolve according to well-known laws, but in many cases microscopic details of motion are not fully accessible. Due to the large system size these influences on the dynamics can be approximated as effective random noise with a certain postulated distribution. The actual origin of the noise, which may be related to chaotic motion (see Chapter 2) or thermal interactions (see Chapter 4), is usually ignored.

Type
Chapter
Information
Complexity Science
The Warwick Master's Course
 , pp. 125 - 209
Publisher: Cambridge University Press
Print publication year: 2013

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