随快速起伏高斯过程演化的随机标量场的相关函数
Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process
摘要：
我们考虑一个由平流-扩散方程（或更一般的发展方程）控制的具有快速起伏的高斯分布随机系数的标量场。在白噪声极限下，我们用三种不同的策略导出了随机标量场系综平均的封闭发展方程，Feynman-Kac公式，Ornstein-Uhlenbeck过程的极限，以及计算传播子在$n$ -单纯形上的簇展开。本文从系综平均的发展方程出发，研究了两种不同类型的流（随机周期流和随机应变流）的被动标量输运问题。对于周期性流动，利用均匀化方法，我们证明了随机标量场的$N$点相关函数在长时间内满足有效的扩散方程。对于应变流，我们显式地计算了随机标量场的平均值，并证明了随机标量场的统计量与几何布朗运动的时间积分有联系。有趣的是，所有归一化矩（例如，偏斜度，峰度标量场在长时间内发散，这意味着标量在其衰减期间变得越来越间歇。
英文摘要：
We consider a scalar field governed by an advection-diffusion equation (or a more general evolution equation) with rapidly fluctuating, Gaussian distributed random coefficients. In the white noise limit, we derive the closed evolution equation for the ensemble average of the random scalar field by three different strategies, i.e., Feynman-Kac formula, the limit of Ornstein-Uhlenbeck process, and evaluating the cluster expansion of the propagator on an $n$-simplex. With the evolution equation of ensemble average, we study the passive scalar transport problem with two different types of flows, a random periodic flow, and a random strain flow. For periodic flows, by utilizing the homogenization method, we show that the $N$-point correlation function of the random scalar field satisfies an effective diffusion equation at long times. For the strain flow, we explicit compute the mean of the random scalar field and show that the statistics of the random scalar field have a connection to the time integral of geometric Brownian motion. Interestingly, all normalized moment (e.g., skewness, kurtosis) of this random scalar field diverges at long times, meaning that the scalar becomes more and more intermittent during its decay.
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