en>Monkbot: /* Further reading */Fix CS1 deprecated date parameter errors

2014-01-29T03:17:56Z

Further reading: Fix CS1 deprecated date parameter errors

New page

In [[probability theory]], an '''interacting particle system''' (IPS) is a [[stochastic process]] <math> (X(t))_{t \in \mathbb R^+} </math> on some configuration space <math> \Omega= S^G </math> given by a site space, a [[Countable infinite|countable-infinite]] [[graph (mathematics)|graph]] <math> G </math> and a local state space, a [[Compact space|compact]] [[metric space]] <math> S </math>. More precisely IPS are continuous-time [[Markov process|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 (mathematics)|contact process]], the [[asymmetric simple exclusion process]] (ASEP), the [[Glauberdynamics]] and in particular the stochastic [[Ising model]].

IPS are usually defined via their [[Infinitesimal generator (stochastic processes)|Markov generator]] giving rise a unique [[Markov process]] using Markov [[semigroups]] and the [[Hille-Yosida theorem]]. The generator again is given via so-called transition rates <math>c_\Lambda(\eta,\xi)>0</math> where <math>\Lambda\subset G</math> is a finite set of sites and <math>\eta,\xi\in\Omega</math> with <math>\eta_i=\xi_i</math> for all <math>i\notin\Lambda</math>. The rates describe exponential waiting times of the process to jump from configuration <math>\eta</math> into configuration <math>\xi</math>. More generally the transition rates are given in form of a finite measure <math>c_\Lambda(\eta,d\xi)</math> on <math>S^\Lambda</math>. The generator <math>L</math> of an IPS has the following form: Let <math>f</math> be an observable in the domain of <math>L</math> which is a subset of the real valued [[continuous function]] on the configuration space, then

<math>Lf(\eta)=\sum_\Lambda\int_{\xi:\xi_{\Lambda^c}=\eta_{\Lambda^c}}c_\Lambda(\eta,d\xi)[f(\xi)-f(\eta)]</math>.

For example for the stochastic [[Ising model]] we have <math>G=\mathbb Z^d</math>, <math>S=\{-1,+1\}</math>, <math>c_\Lambda=0</math> if <math>\Lambda\neq\{i\}</math> for some <math>i\in G</math> and

<math>c_i(\eta,\eta^i)=\exp[-\beta\sum_{j:|j-i|=1}\eta_i\eta_j]</math>

where <math>\eta^i</math> is the configuration equal to <math>\eta</math> except it is flipped at site <math>i</math>. <math>\beta</math> is a new parameter modeling the inverse temperature.

== References ==
*{{cite journal
|last=Liggett
|first=Thomas M.
|authorlink=Thomas M. Liggett
|title=Stochastic Models of Interacting Systems
|year=1997
|journal=The Annals of Probability
|volume=25
|issue=1
|pages=1–29
|issn=00911798
|publisher=Institute of Mathematical Statistics
|doi=10.2307/2959527}}
*{{cite book
|last=Liggett
|first=Thomas M.
|authorlink=Thomas M. Liggett
|title=Interacting Particle Systems
|year=1985
|publisher=Springer Verlag
|location=New York
|isbn=0-387-96069-4 }}

[[Category:Stochastic processes]]
[[Category:Lattice models]]
[[Category:Markov processes]]
[[Category:Self-organization]]
[[Category:Complex systems theory]]
[[Category:Spatial processes]]
[[Category:Stochastic models]]
[[Category:Markov models]]

Rank of a partition - Revision history

en>Monkbot: /* Further reading */Fix CS1 deprecated date parameter errors