Thiruthummal Abhiram Anand, Kim Eun-Jin
Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, UK.
Entropy (Basel). 2022 Aug 12;24(8):1113. doi: 10.3390/e24081113.
Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker-Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker-Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.
信息几何是研究和比较非平衡系统的随机微分方程（SDEs）解的有用工具。作为求解福克 - 普朗克方程的一种替代方法，我们提出了一种新方法来计算时间相关的概率密度函数（PDFs），并使用SDEs的蒙特卡罗（MC）模拟来研究信息几何。具体而言，我们开发了一种新的MC SDE方法，以克服计算时间相关PDF和信息几何诊断方面的挑战，并通过利用GPU计算来加速模拟。使用MC SDE模拟，我们重现了从福克 - 普朗克方法中发现的信息几何缩放关系，用于具有线性和立方阻尼项的随机过程的情况。我们通过计算不等时联合PDF展示了MC SDE模拟相对于FPE求解器的优势。对于具有线性阻尼力的线性过程，联合PDF被发现是高斯分布。相比之下，对于具有立方阻尼力的立方过程，即使在稳态下，联合PDF也呈现双峰结构。这表明非线性力会导致有限的记忆时间。此外，还识别并研究了双峰PDF特征中的几种幂律缩放。
