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  • Print publication year: 2013
  • Online publication date: June 2014

3 - Interacting stochastic particle systems

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.

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