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Concept
Interacting particle system
Summary
In probability theory, an interacting particle system (IPS) is a stochastic process on some configuration space given by a site space, a countably-infinite-order graph and a local state space, a compact metric space . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata.
Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model.
IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates where is a finite set of sites and with for all . The rates describe exponential waiting times of the process to jump from configuration into configuration . More generally the transition rates are given in form of a finite measure on .
The generator of an IPS has the following form. First, the domain of is a subset of the space of "observables", that is, the set of real valued continuous functions on the configuration space . Then for any observable in the domain of , one has
For example, for the stochastic Ising model we have , , if for some and
where is the configuration equal to except it is flipped at site . is a new parameter modeling the inverse temperature.
Voter model
The voter model (usually in continuous time, but there are discrete versions as well) is a process similar to the contact process. In this process is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion.
Official source
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