广义Langevin方程扩展Markov参数化计算的模型降阶技术
Model reduction techniques for the computation of extended Markov parameterizations for generalized Langevin equations
摘要：
广义朗之万方程是一个描述粗粒粒子运动的模型，其中耗散力由记忆项表示。这种模型的数值实现需要实现一个随机延迟微分方程和相应的记忆核的估计。在这里，我们开发了一种新的方法来计算一个数据驱动的马尔可夫模型的粒子的运动，他们的速度自相关函数的等距样本。我们的方法绕过了底层内存内核的确定表示它通过约20个辅助变量。该算法是基于一个复杂的变量的Prony方法的指数插值和采用正实引理模型降阶理论提取相关的马尔可夫模型。我们证明了这种方法的潜在的异常扩散的测试情况下，数据分析，然后将我们的方法应用到速度自相关数据的分子动力学模拟的胶体在Lennard-Jones流体。在这两种情况下，VACF和内存内核可以非常准确地复制。此外，我们表明，该算法也可以处理输入数据与大的统计噪声。我们预计，这将是一个非常有用的工具，在未来的研究，涉及动态粗粒化的复杂软物质系统。
英文摘要：
The generalized Langevin equation is a model for the motion of coarse-grained particles where dissipative forces are represented by a memory term. The numerical realization of such a model requires the implementation of a stochastic delay-differential equation and the estimation of a corresponding memory kernel. Here we develop a new approach for computing a data-driven Markov model for the motion of the particles, given equidistant samples of their velocity autocorrelation function. Our method bypasses the determination of the underlying memory kernel by representing it via up to about twenty auxiliary variables. The algorithm is based on a sophisticated variant of the Prony method for exponential interpolation and employs the Positive Real Lemma from model reduction theory to extract the associated Markov model. We demonstrate the potential of this approach for the test case of anomalous diffusion, where data are given analytically, and then apply our method to velocity autocorrelation data of molecular dynamics simulations of a colloid in a Lennard-Jones fluid. In both cases, the VACF and the memory kernel can be reproduced very accurately. Moreover, we show that the algorithm can also handle input data with large statistical noise. We anticipate that it will be a very useful tool in future studies that involve dynamic coarse-graining of complex soft matter systems.
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