稀有事件的数值方法及其在浸润过程中的应用

Many processes in applied sciences are driven by rare but important events. Well-known examples include conformational changes of bio-molecules, chemical reactions, nucleation events during phase transitions, etc. The dynamics proceeds by long waiting periods around metastable states followed by sudden jumps or transitions from one state to another. The main objects of interest are the transition pathways, the transition states and the transition rates. The computation of these quantities represents a major challenge in computational science. The main difficulty arises from the disparate time scales involved in the system. This makes conventional simulation techniques prohibitively expensive. Noticing this difficulty, the PI and co-workers developed a theoretical framework and efficient numerical tools for the study of complex energy landscapes and rare events. These include the zero-temperature string method for computing minimum energy paths, the finite-temperature string method for computing transition tubes, and the minimum action method for computing the most probable transition paths in non-gradient systems. These methods have been successfully applied to many problems from various disciplines, including conformational changes of biomolecules, micro-magnetics, phase transitions of complex fluids, dislocation dynamics in crystalline solids, etc. This project is a continuation of these earlier work. In this project, we will focus on: (1) the development of numerical methods for the computation of transition states in one type of generalized gradient systems; (2) application of the numerical method to study the wetting transition on solid surfaces patterned with micro-structures using the Cahn-Hilliard model; (3) developing numerical methods for the study of transition events in fluctuating hydrodynamics, and apply the method to understand the effect of the velocity field in wetting transition.

应用科学中大量问题都涉及亚稳态之间状态迁移这类稀有事件，如生物分子的构象改变，相变过程中的成核现象等。在此类问题中，主要计算目标是过渡态，迁移路径和迁移速率。由于时间尺度的悬殊，对这类问题的计算极具挑战性。有鉴于此，申请人和合作者发展建立了一套研究稀有事件的理论框架和数值算法，包括计算极小能量路径的零温弦方法，计算迁移管道的有限温度弦方法，以及针对非梯度系统的最小作用力方法。这些方法已成功应用到多个科学领域。本项目是基于申请人之前工作的延续和发展。我们拟研究:(1)计算一类广义梯度系统过渡态的数值方法；(2)将方法应用到具有微观结构固面上的浸润过程，为理解此类固面的超疏水性科学机制提供理论支持；(3)发展针对Fluctuating hydrodynamics模型中状态迁移问题的数值方法，并用该方法量化速度流在浸润过程中的作用。

应用科学中的大量问题均涉及复杂系统的状态迁移过程，包括系统在亚稳态之间的过渡路径，流体力学中的状态迁移过程。另外，移动接触线问题广泛存在于自然界以及人类生活和工业生产过程中，也是流体力学中一类重要问题。本项目围绕复杂系统中稀有事件的算法及应用，流体力学中的过渡路径和移动接触线等方面，对若干问题展开了研究。具体包括：（1）复杂系统中稀有事件算法及其应用的研究； （2）流体力学中状态迁移路径问题的研究；（3）流体力学中移动接触线问题的理论和算法研究。 我们在数学理论和数值方法以及方法应用上均取得了一定的进展, 达到了预期的目标.