论文标题
离散多尺度动力学中的自发随机性和重新归一化组
Spontaneous stochasticity and renormalization group in discrete multi-scale dynamics
论文作者
论文摘要
我们介绍了一类具有离散时间的多尺度系统，这是由于在存在小规模噪声的情况下流体动力学的局限性限制的问题。这些系统是无限维的，并在比例不变的时空晶格上定义。我们提出了一个定性理论，描述了消失的正则化（Inviscid）限制，作为在流图或相应概率内核中作用的重新归一化群体操作员的吸引子。如果吸引子是一种非平凡的概率内核，我们说无粘性极限是自发的随机性：它定义了具有确定性初始和边界条件的随机（马尔可夫）过程求解确定性方程。结果用可解决的模型说明了结果：导致数字湍流的符号系统和扩展相互作用阶段的系统。
We introduce a class of multi-scale systems with discrete time, motivated by the problem of inviscid limit in fluid dynamics in the presence of small-scale noise. These systems are infinite-dimensional and defined on a scale-invariant space-time lattice. We propose a qualitative theory describing the vanishing regularization (inviscid) limit as an attractor of the renormalization group operator acting in the space of flow maps or respective probability kernels. If the attractor is a nontrivial probability kernel, we say that the inviscid limit is spontaneously stochastic: it defines a stochastic (Markov) process solving deterministic equations with deterministic initial and boundary conditions. The results are illustrated with solvable models: symbolic systems leading to digital turbulence and systems of expanding interacting phases.
