Yuan Ruoshi, Wang Xinan, Ma Yian, Yuan Bo, Ao Ping
Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jun;87(6):062109. doi: 10.1103/PhysRevE.87.062109. Epub 2013 Jun 7.
Based on conventional Ito or Stratonovich interpretation, zero-mean multiplicative noise can induce shifts of attractors or even changes of topology to a deterministic dynamics. Such phenomena usually introduce additional complications in analysis of these systems. We employ in this paper a new stochastic interpretation leading to a straightforward consequence: The steady state distribution is Boltzmann-Gibbs type with a potential function severing as a Lyapunov function for the deterministic dynamics. It implies that an attractor corresponds to the local extremum of the distribution function and the probability is equally distributed right on an attractor. We consider a prototype of nonequilibrium processes, noisy limit cycle dynamics. Exact results are obtained for a class of limit cycles, including a van der Pol type oscillator. These results provide a new angle for understanding processes without detailed balance and can be verified by experiments.
基于传统的伊藤或斯特拉托诺维奇解释，零均值乘性噪声可以导致吸引子的偏移，甚至使确定性动力学的拓扑结构发生变化。此类现象通常会给这些系统的分析带来额外的复杂性。在本文中，我们采用一种新的随机解释，得到一个直接的结果：稳态分布是玻尔兹曼 - 吉布斯型的，其势函数作为确定性动力学的李雅普诺夫函数。这意味着一个吸引子对应于分布函数的局部极值，并且概率在吸引子上均匀分布。我们考虑非平衡过程的一个原型，即有噪声的极限环动力学。对于一类极限环，包括范德波尔型振荡器，我们得到了精确结果。这些结果为理解没有细致平衡的过程提供了一个新视角，并且可以通过实验验证。
