Interacting particle system

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, an interacting particle system (IPS) is a <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a> 
 
 
 
 (
 X
 (
 t
 )
 
 )
 
 t
 ∈
 
 
 R
 
 
 +
 
 
 
 
 
 
 {\displaystyle (X(t))_{t\in \mathbb {R} ^{+}}}
 
 on some configuration space 
 
 
 
 Ω
 =
 
 S
 
 G
 
 
 
 
 {\displaystyle \Omega =S^{G}}
 
 given by a site space, a <a href="/facts/Countable_infinite/Wj4DmWPv">countably-infinite-order</a> <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
 and a local state space, a <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Metric_space/gnBBRXtc">metric space</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
. More precisely IPS are continuous-time <a href="/facts/Markov_process/5oP5hntk">Markov jump processes</a> describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of <a href="/facts/Stochastic_cellular_automata/2d0tPcuA">stochastic cellular automata</a>.
Among the main examples are the <a href="/facts/Voter_model/3AxM9HyN">voter model</a>, the <a href="/facts/Contact_process_(mathematics)/KgfajxGv">contact process</a>, the <a href="/facts/Asymmetric_simple_exclusion_process/s9nglk4j">asymmetric simple exclusion process</a> (ASEP), the <a href="/facts/Glauber_dynamics/fRg4wJ9p">Glauber dynamics</a> and in particular the stochastic <a href="/facts/Ising_model/lvBpj2h1">Ising model</a>.
IPS are usually defined via their <a href="/facts/Infinitesimal_generator_(stochastic_processes)/EUHczpU5">Markov generator</a> giving rise to a unique <a href="/facts/Markov_process/5oP5hntk">Markov process</a> using Markov <a href="/facts/Semigroups/rgSdk2kt">semigroups</a> and the <a href="/facts/Hille-Yosida_theorem/s1qSoQuq">Hille-Yosida theorem</a>. The generator again is given via so-called transition rates 
 
 
 
 
 c
 
 Λ
 
 
 (
 η
 ,
 ξ
 )
 >
 0
 
 
 {\displaystyle c_{\Lambda }(\eta ,\xi )>0}
 
 where 
 
 
 
 Λ
 ⊂
 G
 
 
 {\displaystyle \Lambda \subset G}
 
 is a finite set of sites and 
 
 
 
 η
 ,
 ξ
 ∈
 Ω
 
 
 {\displaystyle \eta ,\xi \in \Omega }
 
 with 
 
 
 
 
 η
 
 i
 
 
 =
 
 ξ
 
 i
 
 
 
 
 {\displaystyle \eta _{i}=\xi _{i}}
 
 for all 
 
 
 
 i
 ∉
 Λ
 
 
 {\displaystyle i\notin \Lambda }
 
. The rates describe exponential waiting times of the process to jump from configuration 
 
 
 
 η
 
 
 {\displaystyle \eta }
 
 into configuration 
 
 
 
 ξ
 
 
 {\displaystyle \xi }
 
. More generally the transition rates are given in form of a finite measure 
 
 
 
 
 c
 
 Λ
 
 
 (
 η
 ,
 d
 ξ
 )
 
 
 {\displaystyle c_{\Lambda }(\eta ,d\xi )}
 
 on 
 
 
 
 
 S
 
 Λ
 
 
 
 
 {\displaystyle S^{\Lambda }}
 
.
The generator 
 
 
 
 L
 
 
 {\displaystyle L}
 
 of an IPS has the following form. First, the domain of 
 
 
 
 L
 
 
 {\displaystyle L}
 
 is a subset of the space of "observables", that is, the set of real valued <a href="/facts/Continuous_functions/sKbl02pB">continuous functions</a> on the configuration space 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
. Then for any observable 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in the domain of 
 
 
 
 L
 
 
 {\displaystyle L}
 
, one has

 
 
 
 L
 f
 (
 η
 )
 =
 
 ∑
 
 Λ
 
 
 
 ∫
 
 ξ
 :
 
 ξ
 
 
 Λ
 
 c
 
 
 
 
 =
 
 η
 
 
 Λ
 
 c
 
 
 
 
 
 
 
 c
 
 Λ
 
 
 (
 η
 ,
 d
 ξ
 )
 [
 f
 (
 ξ
 )
 −
 f
 (
 η
 )
 ]
 
 
 {\displaystyle Lf(\eta )=\sum _{\Lambda }\int _{\xi :\xi _{\Lambda ^{c}}=\eta _{\Lambda ^{c}}}c_{\Lambda }(\eta ,d\xi )[f(\xi )-f(\eta )]}
 
.
For example, for the stochastic <a href="/facts/Ising_model/lvBpj2h1">Ising model</a> we have 
 
 
 
 G
 =
 
 
 Z
 
 
 d
 
 
 
 
 {\displaystyle G=\mathbb {Z} ^{d}}
 
, 
 
 
 
 S
 =
 {
 −
 1
 ,
 +
 1
 }
 
 
 {\displaystyle S=\{-1,+1\}}
 
, 
 
 
 
 
 c
 
 Λ
 
 
 =
 0
 
 
 {\displaystyle c_{\Lambda }=0}
 
 if 
 
 
 
 Λ
 ≠
 {
 i
 }
 
 
 {\displaystyle \Lambda \neq \{i\}}
 
 for some 
 
 
 
 i
 ∈
 G
 
 
 {\displaystyle i\in G}
 
 and

c
          
            i
          
        
        (
        η
        ,
        
          η
          
            i
          
        
        )
        =
        exp
        ⁡
        [
        −
        β
        
          ∑
          
            j
            :
            
              |
            
            j
            −
            i
            
              |
            
            =
            1
          
        
        
          η
          
            i
          
        
        
          η
          
            j
          
        
        ]
      
    
    {\displaystyle c_{i}(\eta ,\eta ^{i})=\exp[-\beta \sum _{j:|j-i|=1}\eta _{i}\eta _{j}]}

where 
 
 
 
 
 η
 
 i
 
 
 
 
 {\displaystyle \eta ^{i}}
 
 is the configuration equal to 
 
 
 
 η
 
 
 {\displaystyle \eta }
 
 except it is flipped at site 
 
 
 
 i
 
 
 {\displaystyle i}
 
. 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 is a new parameter modeling the inverse temperature.

Interacting particle system open-in-new

Interacting particle system