ITR: Mesoscopic Modeling and Simulation: A Novel Approach to Monte Carlo Methods

提案#0219211PI：Markos A.KatSoulaki研究所：马萨诸塞大学阿默斯特分校标题：ITR：介观建模和模拟：一种新的蒙特卡罗模拟方法本项目涉及一个基于最近开发的粗粒度随机介观模型的蒙特卡罗模拟的新框架。介观模型是由微观蒙特卡罗(MC)算法借助非平衡统计力学技术严格推导出的渐近极限的随机偏微分方程组。尽管这些模型描述了介观尺度，远远大于基本的分子尺度，但它们仍然包括关于粒子相互作用和动力学的详细微观信息，并可以系统地模拟各向异性和多种微观机制。介观模型的另一个吸引人的特点是包含了直接来自基本主方程的随机波动，并产生了重要的成核、图案形成和选择机制。最后，使用大偏差和WKB展开的形式和严格的渐近性以及初步的数值模拟表明，MC算法和相应的随机介观模型对于成核率和相变等微妙的量产生基本相同的结果。该方案的主要研究目标是：(A)通过使用基于谱的方法数值求解随机介观方程来开发非平衡粗粒度MC算法，并与传统MC进行详细的基准比较；以及(B)将所提出的计算工具应用于先进材料中的图案形成和纳米多孔膜中的分子分离。由于其基本性质和在描述原子与分子之间复杂的非平衡相互作用方面的通用性，分子动力学模拟和蒙特卡罗(MC)算法已成为科学和工程研究的优秀计算工具。随着增强计算能力的出现，这些方法可以为从物理化学和生物过程到生物材料、药物设计、模式识别和图像处理的众多问题提供前所未有的见解。尽管分子算法得到了广泛的应用，相关的计算方法也取得了长足的进步，但分子算法的长度和时间都很短。因此，它们只能在相当短的时间内模拟相对较少的原子或分子。另一方面，在实验中观察到的设备大小和形态特征通常涉及更大的空间和/或时间尺度。应对这种多尺度建模挑战的一个主要障碍是缺乏一个严格的数学和计算框架，该框架提供了从原子尺度到作为微观相互作用结果的复杂的中观和宏观现象的直接联系。在这个方向上，我们的工作集中在开发新的随机模型和算法，能够描述比传统MC模拟更大的长度和时间尺度，同时仍然包含微观细节。我们打算应用计算方法为两个工程问题提供新的见解，这两个问题目前是用传统的MC技术解决的：(1)研究自组织微观机制及其在先进材料图案形成中的作用；(2)分子在纳米多孔膜和膜中的传输和分离。

Proposal # 0219211PI: Markos A. KatsoulakisInstitution: University of Massachusetts AmherstTitle: ITR: Mesoscopic Modeling and Simulation: A Novel Approach to Monte CarloABSTRACTThis project is concerned with a novel framework for Monte Carlo simulations based on recently developed coarse-grained stochastic mesoscopic models. Mesoscopic models are stochastic partial differential equations which are rigorously derived as asymptotic limits from microscopic Monte Carlo (MC) algorithms by means of techniques from non-equilibrium statistical mechanics. Although such models describe the mesoscopic scales, which are much larger than the underlying molecular scales, they still include detailed microscopic information on particle interactions and dynamics and can systematically model anisotropies and multiple micromechanisms. Another attractive feature of mesoscopic models is the inclusion of random fluctuations derived directly from the underlying master equation and yielding important nucleation and pattern formation and selection mechanisms. Finally, formal and rigorous asymptotics using Large Deviation and WKB expansions as well as preliminary numerical simulations indicate that the MC algorithms and the corresponding stochastic mesoscopic models produce essentially identical results for such delicate quantities as nucleation rates and phase transitions. The main research objectives of this proposal are: (a) to develop non-equilibrium coarse-grained MC algorithms by numerically solving the stochastic mesoscopic equations using highly efficient spectral-based methods and carry out detailed benchmarkings against conventional MC, and (b) to apply the proposed computational tools to applications arising in pattern formation in advanced materials and molecular separation in nanoporous films.Because of their fundamental nature and their versatility in describing complex out-of-equilibrium interactions between atoms and molecules, molecular dynamics simulations and Monte Carlo (MC) algorithms have become preeminent computational tools for science and engineering research. With the advent of enhanced computing capabilities, these methods can provide unprecedented insights into numerous problems ranging from physicochemical and biological processes to biomaterials, drug design, pattern recognition, and image processing. Despite their widespread use and the substantial progress in related computational methods, molecular algorithms are limited to short length and time scales. Hence, they are capable of simulating only a relatively small number of atoms or molecules for quite short time periods. On the other hand, device sizes and morphological features observed in experiments often involve much larger spatial and/or temporal scales. A major obstacle in meeting this multiscale modeling challenge is the lack of a rigorous mathematical and computational framework providing a direct link from the atomistic scale to the complex mesoscopic and macroscopic phenomena that are the result of the microscopic interactions. In this direction, our work focuses on developing novel stochastic models and algorithms capable of describing much larger length and time scales than conventional MC simulations while still incorporating microscopic details. We intend to apply the computational methods to provide new insights into two engineering problems, which are currently intractable with conventional MC techniques: (1) the study of self-organizing micromechanisms and their role in pattern formation in advanced materials, and (2) the transport and separation of molecules in nanoporous films and membranes.