In
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 on some configuration space
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 ...
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 ...
. 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
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
.
IPS are usually defined via their
Markov generator giving rise to a unique
Markov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...
using Markov
semigroups
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
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
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
on the configuration space
. Then for any observable
in the domain of
, one has