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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 System size expansion ： ウィキペディア英語版 System size expansion The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen〔van Kampen, N. G. (2007) "Stochastic Processes in Physics and Chemistry", North-Holland Personal Library〕 used in the analysis of stochastic processes. Specifically, it allows one to find an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system. Less formally, it is normally straightforward to write down a mathematical description of a system where processes happen randomly (for example, radioactive atoms randomly decay in a physical system, or genes that are expressed stochastically in a cell). However, these mathematical descriptions are often too difficult to solve for the study of the systems statistics (for example, the mean and variance of the number of atoms or proteins as a function of time). The system size expansion allows one to obtain an approximate statistical description that can be solved much more easily than the master equation. == Preliminaries == Systems that admit a treatment with the system size expansion may be described by a probability distribution $P(X,\; t)$, giving the probability of observing the system in state $X$ at time $t$. $X$ may be, for example, a vector with elements corresponding to the number of molecules of different chemical species in a system. In a system of size $\backslash Omega$ (intuitively interpreted as the volume), we will adopt the following nomenclature: $\backslash mathbf$ is a vector of macroscopic copy numbers, $\backslash mathbf\; =\; \backslash mathbf/\backslash Omega$ is a vector of concentrations, and $\backslash mathbf$ is a vector of deterministic concentrations, as they would appear according to the rate equation in an infinite system. $\backslash mathbf$ and $\backslash mathbf$ are thus quantities subject to stochastic effects. A master equation describes the time evolution of this probability.〔 Henceforth, a system of chemical reactions〔Elf, J. and Ehrenberg, M. (2003) "Fast Evaluation of Fluctuations in Biochemical Networks With the Linear Noise Approximation", ''Genome Research'', 13:2475–2484.〕 will be discussed to provide a concrete example, although the nomenclature of "species" and "reactions" is generalisable. A system involving $N$ species and $R$ reactions can be described with the master equation: :$\backslash frac\; =\; \backslash Omega\; \backslash sum\_^R\; \backslash left(\; \backslash prod\_^\; \backslash mathbb^,\; \backslash Omega)\; P\; (\backslash mathbf,\; t).$ Here, $\backslash Omega$ is the system size, $\backslash mathbb$ is an operator which will be addressed later, $S\_$ is the stoichiometric matrix for the system (in which element $S\_$ gives the stoichiometric coefficient for species $i$ in reaction $j$), and $f\_j$ is the rate of reaction $j$ given a state $\backslash mathbf$ and system size $\backslash Omega$. $\backslash mathbb^$ from the $i$th element of its argument. For example, $\backslash mathbb^,\; x\_3)$. This formalism will be useful later. The above equation can be interpreted as follows. The initial sum on the RHS is over all reactions. For each reaction $j$, the brackets immediately following the sum give two terms. The term with the simple coefficient −1 gives the probability flux away from a given state $\backslash mathbf$ due to reaction $j$ changing the state. The term preceded by the product of step operators gives the probability flux due to reaction $j$ changing a different state $\backslash mathbf$ into state $\backslash mathbf$. The product of step operators constructs this state $\backslash mathbf$. 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「System size expansion」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.