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- The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen 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. (en)
- 系统尺度展开，又称van Kampen's展开或者Ω-展开，是由开创运用于随机过程分析的数学方法。它能够对一个具有非线性变化率系统的主方程的解进行估计。这种展开的第一个项被称为，此时系统主方程的解使用福克-普朗克方程（Fokker-Planck equation）进行估计，其线性估计系数由该系统的变化率和化学计量数决定。 一般来讲，对于一个随机过程系统的数学描述是写下每个变量的微分方程，从而形成微分方程组（例如，在一个物理系统中，描述放射性分子随机衰变，或者在细胞环境中，描述基因的随机表达）。但是这样的微分方程组往往非常复杂，难以得到解析解，进而难以获得关于系统状态的统计量（例如，获取分子数目或者蛋白质数目的平均值或方差随时间变化的方程）。系统尺度展开可以运用统计的方法对这样的复杂的系统进行估计，从而得到该系统的近似解。 (zh)
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- 系统尺度展开，又称van Kampen's展开或者Ω-展开，是由开创运用于随机过程分析的数学方法。它能够对一个具有非线性变化率系统的主方程的解进行估计。这种展开的第一个项被称为，此时系统主方程的解使用福克-普朗克方程（Fokker-Planck equation）进行估计，其线性估计系数由该系统的变化率和化学计量数决定。 一般来讲，对于一个随机过程系统的数学描述是写下每个变量的微分方程，从而形成微分方程组（例如，在一个物理系统中，描述放射性分子随机衰变，或者在细胞环境中，描述基因的随机表达）。但是这样的微分方程组往往非常复杂，难以得到解析解，进而难以获得关于系统状态的统计量（例如，获取分子数目或者蛋白质数目的平均值或方差随时间变化的方程）。系统尺度展开可以运用统计的方法对这样的复杂的系统进行估计，从而得到该系统的近似解。 (zh)
- The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen 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. (en)
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- System size expansion (en)
- 系统尺度展开 (zh)
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