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probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, an interacting particle system (IPS) is a stochastic process (X(t))_ on some configuration space \Omega= S^G given by a site space, a countable-infinite
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
G and a local state space, a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
S . 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 Stochastic cellular automata or probabilistic cellular automata (PCA) or random cellular automata or locally interacting Markov chains are an important extension of cellular automaton. Cellular automata are a discrete-time dynamical system of inte ...
. Among the main examples are the
voter model In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975. One can imagine that there is a "voter" at each point on a connected graph, where the ...
, the
contact process The contact process is the current method of producing sulfuric acid in the high concentrations needed for industrial processes. Platinum was originally used as the catalyst for this reaction; however, as it is susceptible to reacting with arseni ...
, the
asymmetric simple exclusion process In probability theory, the asymmetric simple exclusion process (ASEP) is an interacting particle system introduced in 1970 by Frank Spitzer. Many articles have been published on it in the physics and mathematics literature since then, and it has ...
(ASEP), the
Glauber dynamics In statistical physics, Glauber dynamics is a way to simulate the Ising model (a model of magnetism) on a computer. It is a type of Markov Chain Monte Carlo algorithm. The algorithm In the Ising model, we have say N particles that can spin u ...
and in particular the stochastic Ising model. IPS are usually defined via their Infinitesimal generator (stochastic processes), 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 c_\Lambda(\eta,\xi)>0 where \Lambda\subset G is a finite set of sites and \eta,\xi\in\Omega with \eta_i=\xi_i for all i\notin\Lambda. The rates describe exponential waiting times of the process to jump from configuration \eta into configuration \xi. More generally the transition rates are given in form of a finite measure c_\Lambda(\eta,d\xi) on S^\Lambda. The generator L of an IPS has the following form. First, the domain of L is a subset of the space of "observables", that is, the set of real valued continuous functions on the configuration space \Omega. Then for any observable f in the domain of L, one has Lf(\eta)=\sum_\Lambda\int_c_\Lambda(\eta,d\xi)[f(\xi)-f(\eta)]. For example, for the stochastic Ising model we have G=\mathbb Z^d, S=\, c_\Lambda=0 if \Lambda\neq\ for some i\in G and :c_i(\eta,\eta^i)=\exp[-\beta\sum_\eta_i\eta_j] where \eta^i is the configuration equal to \eta except it is flipped at site i. \beta is a new parameter modeling the inverse temperature.


The Voter model

The
voter model In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975. One can imagine that there is a "voter" at each point on a connected graph, where the ...
(usually in continuous time, but there are discrete versions as well) is a process similar to the contact process(mathematics), contact process. In this process \eta(x) 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. One can generalize the process by allowing the picking of neighbors to be something other than uniform.


Discrete time process

In the discrete time voter model in one dimension, \xi_t(x): \mathbb \to \ represents the state of particle x at time t. Informally each individual is arranged on a line and can "see" other individuals that are within a radius, r. If more than a certain proportion, \theta of these people disagree then the individual changes her attitude, otherwise she keeps it the same. Rick Durrett, Durrett and Steif (1993) and Steif (1994) show that for large radii there is a critical value \theta_c such that if \theta > \theta_c most individuals never change, and for \theta \in (1/2, \theta_c) in the limit most sites agree. (Both of these results assume the probability of \xi_0(x) = 1 is one half.) This process has a natural generalization to more dimensions, some results for this are discussed in Rick Durrett, Durrett and Steif (1993).


Continuous time process

The continuous time process is similar in that it imagines each individual has a belief at a time and changes it based on the attitudes of its neighbors. The process is described informally by Thomas M. Liggett, Liggett (1985, 226), "Periodically (i.e., at independent exponential times), an individual reassesses his view in a rather simple way: he chooses a 'friend' at random with certain probabilities and adopts his position." A model was constructed with this interpretation by Holley and Thomas M. Liggett, Liggett (1975). This process is equivalent to a process first suggested by Clifford and Sudbury (1973) where animals are in conflict over territory and are equally matched. A site is selected to be invaded by a neighbor at a given time.


References

* * * * * * {{Stochastic processes Lattice models Self-organization Complex systems theory Spatial processes Markov models