无细致平衡非平衡过程的随机动力学结构（SDS）。II：具有非线性力和乘性噪声的SDS的构造
Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. II: construction of SDS with nonlinear force and multiplicative noise
摘要：
非平衡行为中的涌现现象可以用一组随机微分方程很好地描述。受现代生物学中噬菌体λ基因开关的鲁棒性和稳定性研究的启发，我们发现存在一种通用非平衡过程的分类：在随机微分方程的连续描述中，存在四个动力学元素：势函数$ϕ$，摩擦矩阵$ S$，反对称矩阵$ T $和噪声。缺乏细致平衡的一般特征则由$T$精确地表示。对于不动点附近的动力系统，无论其是否稳定，其随机动力系统都是线性的。已经进行了相当完整的分析（Kwon，Ao，Jumless，cond-mat/0506280; PNAS，{ \bf 102}（2005）13029），称为SDS I。一个重要的和持久的问题是存在的势函数与非线性力和乘性噪声，具有良好的局部动力学和全局定态性质。在这里，我们证明了一个动态结构内置到随机微分方程允许我们构建这样一个全局优化势函数。首先，我们提供建筑。其中最重要的组成部分是广义爱因斯坦关系。然后，我们提出了一个近似方案：梯度展开，把每一个订单成线性矩阵方程。这种方法与其他已知的随机处理的一致性将在下一篇论文SDS III中讨论;与统计力学和热力学的明确联系将在即将发表的论文SDS IV中讨论。
英文摘要：
There is a whole range of emergent phenomena in non-equilibrium behaviors can be well described by a set of stochastic differential equations. Inspired by an insight gained during our study of robustness and stability in phage lambda genetic switch in modern biology, we found that there exists a classification of generic nonequilibrium processes: In the continuous description in terms of stochastic differential equations, there exists four dynamical elements: the potential function $ϕ$, the friction matrix $ S$, the anti-symmetric matrix $ T $, and the noise. The generic feature of absence of detailed balance is then precisely represented by $T$. For dynamical near a fixed point, whether or not it is stable or not, the stochastic dynamics is linear. A rather complete analysis has been carried out (Kwon, Ao, Thouless, cond-mat/0506280; PNAS, {\bf 102} (2005) 13029), referred to as SDS I. One important and persistent question is the existence of a potential function with nonlinear force and with multiplicative noise, with both nice local dynamical and global steady state properties. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. First, we provide the construction. One of most important ingredient is the generalized Einstein relation. We then present an approximation scheme: The gradient expansion which turns every order into linear matrix equations. The consistent of such methodology with other known stochastic treatments will be discussed in next paper, SDS III; and the explicitly connection to statistical mechanics and thermodynamics will be discussed in a forthcoming paper, SDS IV.
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