Analysis of multi-scale problems in mathematical chemistry

在过去的几十年里，人们对微米和纳米尺度的科学和工程产生了极大的兴趣，从而产生了诸如纳米技术和生物分子建模等全新领域。因此，分子动力学作为统计力学的一个分支，起源于20世纪50年代，已成为化学、物理和生物等许多应用科学领域的关键技术，也是材料科学和工程中许多应用的新兴方法。MD对数学家提出了令人着迷的挑战，因为系统是大的、复杂的、多尺度的，特别是离散的。对离散模型的系统研究还处于起步阶段。一个由数学家和科学家组成的巴斯团队已经确定了三个我们想要详细理解的问题：(i)连接蛋白质能量景观最小值的最小能量路径：化学和生物学中的许多系统可以获得几种构型。从一种配置到另一种配置的变化是罕见的，但通常是重要的事件。我们如何以可靠的方式计算相应的轨迹？（ii）动态相变：轨迹空间中的相变是一个引人入胜的新课题，它是由玻璃材料驱动的。我们的目标是朝着这种时空相变的数学理论迈出第一步。（iii） NiTi中马氏体相变的多尺度建模：许多材料和物质都是通过高度复杂的能量景观来建模的。虽然第一个主题旨在理解这种景观中的轨迹，但我们也试图用从头计算的数据对一种特定材料的景观进行建模。这项活动将促进英国物理化学中多尺度问题的新兴数学研究，探索新的科学领域并为后续应用奠定基础，并使MD的用户接触到最先进的数学技术。我们设想这项活动将成为科学家和数学家之间长期双向互动的核心点。我们的愿景是，数学的贡献可能会导致MD模拟中更强大、更有效的技术，而来自科学的输入可以作为复杂系统的范例，并可能刺激处理复杂性的新数学工具的发展。例如，DNA动力学分析是计算哈密顿轨迹的数学算法的一个具有挑战性的测试案例。我们的目标是将巴斯最近开发的方法应用于这样一个测试案例，即蛋白质G(9)，尽管它相对较小（56个残基），但它包含蛋白质的典型二级结构元素，一个α -螺旋和两个β -链。我们相信，这种跨学科的互动将在医学博士领域取得丰硕的成果，医学博士将成为本世纪的主导技术。

The past decades have witnessed an explosion of interest into science and engineering at micrometre and nanometre scales, which has given rise to whole new fields such as nanotechnology and biomolecular modelling. As a consequence molecular dynamics (MD), which originated in the 1950s as a branch of statistical mechanics, has become a key technique in many areas of applied science, including chemistry, physics and biology, and is also an emerging method in many applications in materials science and engineering. MD poses fascinating challenges to mathematicians, since the systems are large, complex, multiscale, and in particular discrete. Systematic research on discrete models is still in its infancy. A Bath team of mathematicians and scientists has identified three problems we would like to understand in great detail: (i) Minimal energy path connecting the minima of a protein energy landscape: Many systems in chemistry and biology can attain several configurations. A change from one configuration to another is a rare but normally important event. How can we calculate the corresponding trajectories in a reliable way? (ii) Dynamic phase transitions: A fascinating new topic are phase transitions in trajectory space, which are motivated by glassy materials. We aim to develop the first steps toward a mathematical theory of such space-time phase transitions. (iii) Multiscale modelling of martensitic phase transitions in NiTi: Many materials and substances are modelled by highly complex energy landscapes. While the first topic aims at understanding trajectories in such a landscape, we also seek to model the landscape for one particular material with data from ab initio calculations. This activity will - foster emerging mathematical research on multiscale problems in physical chemistry in the UK, - explore new scientific areas and prepare the ground for subsequent applications,- and expose users of MD to state-of-the-art mathematical techniques. We envision this activity as a nucleation point for a long-term two-way interaction between scientists and mathematicians. The vision is that mathematical contributions potentially lead to more robust and efficient techniques in MD simulations, while the input from the Sciences serves as a paradigm for complex systems and may stimulate the development of new mathematical tools dealing with complexity. For example, the analysis of DNA dynamics is a challenging test case for mathematical algorithms to compute Hamiltonian trajectories. We aim to apply methods recently developed in Bath to such a test case, namely the protein G (9) which, despite being relatively small (56 residues), nevertheless includes typical secondary structure elements of proteins, one alpha-helix and two beta-strands. We believe that such interdisciplinary interaction will be very fruitful in MD, which will be a dominant technology in this century.