With the growing number of microscale devices from computer memory to microelectromechanical systems, such as lab-on-a-chip biosensors and the increased ability to experimentally measure at the micro- and nanoscale, modeling systems with stochastic processes is a growing need across science. In particular, stochastic partial differential equations (SPDEs) naturally arise from continuum models—for example, a pillar magnet's magnetization or an elastic membrane's mechanical deflection. In this review, I seek to acquaint the reader with SPDEs from the point of view of numerically simulating their finite-difference approximations, without the rigorous mathematical details of assigning probability measures to the random field solutions. I will stress that these simulations with spatially uncorrelated noise may not converge as the grid size goes to zero in the way that one expects from deterministic convergence of numerical schemes in two or more spatial dimensions. I then present some models with spatially correlated noise that maintain sampling of the physically relevant equilibrium distribution. Numerical simulations are presented to demonstrate the dynamics; the code is publicly available on GitHub.

中文翻译：

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具有空间相关噪声的随机偏微分方程入门


随着从计算机内存到微机电系统（如芯片实验室生物传感器）的微尺度设备数量的增加，以及在微米和纳米尺度上进行实验测量的能力的提高，具有随机过程的建模系统是整个科学领域日益增长的需求。特别是，随机偏微分方程 （SPDE） 自然产生于连续体模型，例如，支柱磁体的磁化或弹性膜的机械偏转。在这篇评论中，我试图从数值模拟有限差分近似的角度让读者熟悉 SPDE，而无需为随机场解分配概率测度的严格数学细节。我将强调，这些具有空间不相关噪声的模拟可能不会收敛，因为网格大小会像人们期望的两个或多个空间维度中数值方案的确定性收敛那样归零。然后，我提出了一些具有空间相关噪声的模型，这些模型保持了物理相关平衡分布的采样。提出了数值模拟以证明动力学;代码在 GitHub 上公开提供。