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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Contact process (mathematics) ： ウィキペディア英語版 Contact process (mathematics) The contact process is a model of an interacting particle system. It is a continuous time Markov process with state space $\backslash ^S$, where $S$ is a finite or countable graph, usually Z$\{\}^d$. The process is usually interpreted as a model for the spread of an infection: if the state of the process at a given time is $\backslash eta$, then a site $x$ in $S$ is "infected" if $\backslash eta(x)=1$ and healthy if $\backslash eta(x)=0$. Infected sites become healthy at a constant rate, while healthy sites become infected at a rate proportional to the number infected neighbors. One can generalize the state space to $\backslash \{0,\backslash ldots,\; \backslash kappa\backslash \}^S$, such is called the multitype contact process. It represents a model when more than one type of infection is competing for space. ==Dynamics== More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site $x$, :$1\backslash rightarrow0\backslash quad\backslash mbox1,$ :$0\backslash rightarrow1\backslash quad\backslash mbox\backslash lambda\backslash sum\_\backslash eta(y),$ where the sum is over all the neighbors in $S$ of $x$. This means that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and vice versa). For each graph $S$ there exists a critical value $\backslash lambda\_c$ for the parameter $\backslash lambda$ so that if $\backslash lambda>\backslash lambda\_c$ then the 1's survive (that is, if there is at least one 1 at time zero, then at any time there are ones) with positive probability, while if then the process dies out. For contact process on the integer lattice, a major breakthrough came in 1990 when Bezuidenhout and Grimmett showed that the contact process also dies out at the critical value. Their proof makes use of eded|date=June 2012}} came in 1990 when Bezuidenhout and Grimmett showed that the contact process also dies out at the critical value. Their proof makes use of percolation theory. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Contact process (mathematics)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.