SIAM Review, Volume 62, Issue 4, Page 901-935, January 2020.
We study the time correlation functions of coupled linear Langevin dynamics with and without inertial effects, both analytically and numerically. The model equation represents the physical behavior of a harmonic oscillator in two or three dimensions in the presence of friction, additive noise, and an external field with both rotational and deformational components. This simple model plays pivotal roles in understanding more complicated processes. The analytical solution presented serves as a test of numerical integration schemes, and its derivation is presented in a fashion that allows it to be repeated directly in a classroom. While the results in the absence of fields (equilibrium) or confinement (free particle) are omnipresent in the literature, we write down, apparently for the first time, the full nonequilibrium results that may correspond, e.g., to a Hookean dumbbell embedded in a macroscopically homogeneous shear or mixed flow field. We demonstrate how the inertial results reduce to their noninertial counterparts in the nontrivial limit of vanishing mass. While the results are derived using basic integrations over Dirac delta distributions, we also provide alternative approaches involving (i) Fourier transforms, which seem advantageous only if the measured quantities also reside in Fourier space, and (ii) a Fokker--Planck equation and the moments of the probability distribution. The results, verified by numerical experiments, provide additional means of measuring the performance of numerical methods for such systems. It should be emphasized that this article provides specific details regarding the derivations of the time correlation functions as well as the implementations of various numerical methods, so that it can serve as a standalone piece for lessons in the framework of Itô stochastic differential equations and calculus.

中文翻译：

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平衡和非平衡兰格文动力学的时间相关函数：使用随机数的导数和数值

SIAM评论，第62卷，第4期，第901-935页，2020年1月。
我们在分析和数值上研究了带有和不带有惯性效应的耦合线性兰格文动力学的时间相关函数。模型方程式在存在摩擦，附加噪声以及具有旋转和变形分量的外部场的情况下，以二维或三维形式表示谐波振荡器的物理行为。这个简单的模型在理解更复杂的过程中起着关键作用。所提供的分析解决方案可作为数值积分方案的测试，其派生方式可以直接在教室中重复进行。虽然没有场（平衡）或约束（自由粒子）的结果在文献中不存在，但我们显然是第一次写下了可能对应的全部非平衡结果，例如，嵌入在宏观均匀剪切或混合流场中的Hookean哑铃。我们证明了惯性结果如何在消失质量的非平凡极限中减少到其非惯性对应物。虽然结果是使用Dirac三角洲分布的基本积分得出的，但我们还提供了（i）傅立叶变换的替代方法，仅当测得的量也位于傅立叶空间中时，这才是有利的；以及（ii）Fokker-Planck方程概率分布的时刻。通过数值实验验证的结果为测量此类系统的数值方法性能提供了其他手段。