Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom.
Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom.
Phys Rev E. 2019 Jan;99(1-1):013303. doi: 10.1103/PhysRevE.99.013303.
Molecular dynamics represents a key enabling technology for applications ranging from biology to the development of new materials. However, many real-world applications remain inaccessible to fully resolved simulations due to their unsustainable computational costs and must therefore rely on semiempirical coarse-grained models. Significant efforts have been devoted in the last decade towards improving the predictivity of these coarse-grained models and providing a rigorous justification of their use, through a combination of theoretical studies and data-driven approaches. One of the most promising research efforts is the (re)discovery of the Mori-Zwanzig projection as a generic, yet systematic, theoretical tool for deriving coarse-grained models. Despite its clean mathematical formulation and generality, there are still many open questions about its applicability and assumptions. In this work, we propose a detailed derivation of a hybrid multiscale system, generalizing and further investigating the approach developed in Español [Europhys. Lett. 88, 40008 (2009)10.1209/0295-5075/88/40008]. Issues such as the general coexistence of atoms (fully resolved degrees of freedom) and beads (larger coarse-grained units), the role of the fine-to-coarse mapping chosen, and the approximation of effective potentials are discussed. The theoretical discussion is supported by numerical simulations of a monodimensional nonlinear periodic benchmark system with an open-source parallel Julia code, easily extensible to arbitrary potential models and fine-to-coarse mapping functions. The results presented highlight the importance of introducing, in the macroscopic model, nonconstant fluctuating and dissipative terms, given by the Mori-Zwanzig approach, to correctly reproduce the reference fine-grained results, without requiring ad hoc calibration of interaction potentials and thermostats.
分子动力学是生物学到新材料开发等众多应用领域的关键使能技术。然而，由于其不可持续的计算成本，许多实际应用仍然无法进行完全解析的模拟，因此必须依赖于半经验的粗粒化模型。在过去的十年中，人们做出了巨大的努力，通过理论研究和数据驱动方法的结合，来提高这些粗粒化模型的预测能力，并为其使用提供严格的依据。最有前途的研究工作之一是重新发现 Mori-Zwanzig 投影作为一种通用的、系统的理论工具，用于推导粗粒化模型。尽管它具有干净的数学公式和通用性，但关于其适用性和假设仍然存在许多悬而未决的问题。在这项工作中，我们提出了一个混合多尺度系统的详细推导，推广和进一步研究了在 Español [Europhys. Lett. 88, 40008 (2009)10.1209/0295-5075/88/40008]中开发的方法。讨论了诸如原子（完全解析自由度）和珠粒（更大的粗粒化单元）的普遍共存、选择的细-粗映射的作用以及有效势的近似等问题。理论讨论得到了一个具有开源并行 Julia 代码的一维非线性周期基准系统的数值模拟的支持，该代码易于扩展到任意的势模型和细-粗映射函数。所提出的结果强调了在宏观模型中引入 Mori-Zwanzig 方法给出的非恒定波动和耗散项的重要性，以正确再现参考细粒度结果，而无需对相互作用势和热库进行特定的校准。
