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

提案编号0219211PI: Markos A. KatsoulakisInstitution: University of Massachusetts amherst标题：ITR: Mesoscopic Modeling and Simulation：一种新的Monte Carlo方法摘要该项目涉及基于最近开发的粗粒度随机Mesoscopic模型的Monte Carlo模拟的新框架。介观模型是一种随机偏微分方程，它是用非平衡统计力学的方法从微观蒙特卡罗算法中严格导出的渐近极限。虽然这些模型描述的是比分子尺度大得多的介观尺度，但它们仍然包含了关于粒子相互作用和动力学的详细微观信息，并且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从潜在的主方程推导出来的随机波动，并给出了重要的成核和模式形成和选择机制。最后，使用大偏差和WKB展开的形式和严格渐近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成核速率和相变等精细量产生了基本相同的结果。本方案的主要研究目标是：(a)通过使用高效的基于光谱的方法数值求解随机介观方程，开发非平衡粗粒度MC算法，并对传统MC进行详细的基准测试；(b)将所提出的计算工具应用于先进材料的模式形成和纳米孔膜的分子分离。由于分子动力学模拟和蒙特卡罗（MC）算法的基本性质及其在描述原子和分子之间复杂的非平衡相互作用方面的通用性，它们已成为科学和工程研究的卓越计算工具。随着增强的计算能力的出现，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等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，但它仍然局限于较短的长度和时间尺度。因此，它们只能在相当短的时间内模拟相对少量的原子或分子。另一方面，实验中观察到的器件尺寸和形态特征往往涉及更大的空间和/或时间尺度。应对这一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，提供从原子尺度到复杂的介观和宏观现象的直接联系，这些现象是微观相互作用的结果。在这个方向上，我们的工作重点是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目前传统MC技术难以解决的两个工程问题提供新的见解：(1)自组织微机制及其在先进材料中模式形成中的作用的研究；(2)纳米孔膜和膜中分子的运输和分离。

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.