Multiscale Modeling &Simulation, Volume 18, Issue 2, Page 1113-1135, January 2020.
The Mori--Zwanzig formalism is applied to derive an equation for the evolution of linear observables of the overdamped Langevin equation. To illustrate the resulting equation and its use in deriving approximate models, a particular benchmark example is studied both numerically and via a formal asymptotic expansion. The example considered demonstrates the importance of memory effects in determining the correct temporal behavior of such systems.

中文翻译：

---

通过Mori-Zwanzig形式主义对过度阻尼的Langevin动力学进行粗粒度处理

《多尺度建模与仿真》，第18卷，第2期，第1113-1135页，2020年1月。
采用Mori-Zwanzig形式主义来推导过度阻尼的Langevin方程的线性可观现象的演化方程。为了说明所得方程及其在推导近似模型中的用途，对一个特定的基准示例进行了数值研究，并通过形式渐近展开进行了研究。所考虑的示例证明了记忆效应在确定此类系统的正确时间行为方面的重要性。