Structure of stochastic dynamics near fixed points
Structure of stochastic dynamics near fixed points
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DOI：
10.1073/pnas.0506347102
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发表时间：
2005-09-13
影响因子：
11.1
通讯作者：
Thouless, DJ
中科院分区：
文献类型：
--
作者：
Kwon, C;Ao, P;Thouless, DJ
我们分析了随机动力学的结构附近的稳定或不稳定的固定点，力可以近似线性化。我们发现，一个成本函数，确定玻尔兹曼平稳分布总是可以定义在它附近。这样的平稳分布不需要满足通常的详细平衡条件，但可能会有一个发散的概率流。在线性的情况下，力可以分为两部分，其中一部分给出了详细的平衡与扩散运动，而另一个引起循环运动的表面上的恒定成本函数。通过使用力矩阵的Jordan变换，我们找到了成本函数的显式构造。我们讨论的奇异性的变换及其后果的平稳分布。这种玻尔兹曼分布可能不是唯一的，非线性效应和边界条件可能会改变分布，甚至在固定点附近也会引起额外的电流。
We analyze the structure of stochastic dynamics near either a stable or unstable fixed point, where the force can be approximated by linearization. We find that a cost function that determines a Boltzmann-like stationary distribution can always be defined near it. Such a stationary distribution does not need to satisfy the usual detailed balance condition but might have instead a divergence-free probability current. In the linear case, the force can be split into two parts, one of which gives detailed balance with the diffusive motion, whereas the other induces cyclic motion on surfaces of constant cost function. By using the Jordan transformation for the force matrix, we find an explicit construction of the cost function. We discuss singularities of the transformation and their consequences for the stationary distribution. This Boltzmann-like distribution may be not unique, and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point.