Brownian bridges for stochastic problems in chemical sciences

连续随机游动--随时间漂移的噪声过程--在广泛的化学领域中无处不在，例如在研究聚合物分子动力学、识别化学反应路径以及在分子尺度上量化热量和质量传递过程中。在许多实际情况下，人们感兴趣的是检查其路径开始和结束于代表不同化学物种或分子构型的特定值集内的随机游动。这样的想法将使人们能够量化化学过程中的罕见事件，或者反过来，量化最可能的构型和反应路径。系统地产生这些随机游动的一个很有前途的方法是利用一个被称为随机桥的概念，这个概念已经被用于化学过程控制理论，以引导噪声过程到达安全和有利可图的最终状态。然而，尽管这一想法在聚合物物理和分子模拟中具有广泛的应用潜力，但它尚未适应于化学应用。这一建议为两个化学应用程序制定了随机桥梁，在这两个应用程序中，传统的解释噪声的技术在计算上效率低下。第一个应用考虑了特定构型的半柔性聚合物链的生长，这在研究DNA分子在生化过程中的行为时很有用。第二个应用重点是有效地检测结晶过程中的不同反应路径，这对制造药物至关重要。研究人员将通过国家会议传播他们的成果，并将向当地一家围绕随机行走主题的科学博物馆提供科学演示。为了通过计算研究由连续随机行走控制的动力学的广泛分子尺度化学过程，研究团队将追求两个主要研究目标。第一个目标是发展高效的随机桥的数值技术，使连续的随机游动在指定的区域结束，停留在指定的区域，或在一个区域之前到达另一个区域。目前，所有的随机桥都是通过求解向后福克-普朗克(BFP)方程和偏微分方程(PDE)来建立的，然后使用该解来计算引导路径到达期望相空间区域的有效漂移。这种方法背后最大的障碍是当人们不能很容易地准确地计算PDE解时，这是复杂或高维系统的情况。第一个目标将利用BFP方程的渐近性质来产生一个近似漂移，以引导随机游走到相空间的正确区域，并确定校正(即重新加权)该近似所产生的任何误差的方法。这一过程将创建一种计算效率高的方法来生成条件随机游动，这种游动可以扩展到更高的维度，并可以用于复杂的分子系统。第二个目标将在两个应用中使用这些想法。第一个例子将侧重于模拟连续高分子链在外场中的动力学--这是所有高分子场理论中的一个典型问题。所提出的研究计划将证明，随机桥可以有效地采样以给定拓扑结束或以一定范围的最终能量结束的聚合物构象，后者对于罕见事件采样是重要的。第二个例子将考虑结晶过程中的成核路径，并将显示桥可以有效地对自由能景观中的路径进行采样，这些路径先达到一种晶体构象(晶型)，最终可以控制晶体产品多态之间的选择性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Continuous random walks – noisy processes that drift in time – are omnipresent in a wide range of chemical fields such as in studying polymer molecule dynamics, identifying chemical reaction pathways, and quantifying heat and mass transfer processes at the molecular scale. In many practical situations, one is interested in examining random walks whose paths start and end within specified sets of values that represent distinct chemical species or molecule configurations. Such ideas would allow one to quantify rare events in a chemical process, or conversely, quantify the most probable configurations and reaction pathways. One promising methodology to systematically generate these random walks is to make use of a concept known as a stochastic bridge, an idea that has been used in chemical process control theory to guide noisy processes to safe and profitable end states. However, this idea has yet to be adapted to applications in chemistry despite its potential for a wide range of applications in polymer physics and molecular simulations. This proposal formulates stochastic bridges for two chemistry applications where traditional techniques to account for the noise are computationally inefficient. The first application considers the growth of semi-flexible polymer chains of a specific configuration, which is useful when studying DNA molecule behavior during biochemical processes. The second application focuses on efficiently examining different reaction pathways during crystallization, which is vital for manufacturing pharmaceuticals. The investigators will disseminate their results through national conferences and will provide science demonstrations to a local science museum that will revolve around the theme of random walks.To computationally study the wide range of molecular-scale chemical processes with dynamics governed by continuous random walks, two primary research objectives will be pursued by the research team. The first objective is to develop highly efficient numerical techniques for stochastic bridges that condition continuous random walks to end in a specified region, stay in a given region, or reach one region before another. Currently, all stochastic bridges are created by solving a Backwards Fokker Planck (BFP) equation, a partial differential equation (PDE), and then using this solution to compute an effective drift that guides paths towards the desired region of phase space. The largest roadblock behind this approach occurs when one cannot readily compute the PDE solution exactly, which is the case for complex or high dimensional systems. The first objective will use the asymptotic properties of the BFP equation to generate an approximate drift that guides random walks to the correct regions of phase space, and determine ways to correct (i.e., re-weight) any errors incurred from this approximation. This procedure will create a computationally efficient method to generate conditioned random walks that can scale to higher dimensions and can be used for complex molecular systems. The second objective will employ these ideas in two applications. The first example will focus on simulating the dynamics of a continuous polymer chain in an external field - a canonical problem in all polymer field theories. The proposed research program will demonstrate that a stochastic bridge can effectively sample polymer conformations that end with a given topology or end with a range of final energies, the latter of which is important for rare event sampling. The second example will consider nucleation pathways during crystallization and will show that a bridge can efficiently sample paths in a free energy landscape that reach one crystal conformation (polymorph) before others, ultimately making it possible to control selectivity between crystal product polymorphs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.