Analysis of multi-scale problems in mathematical chemistry

数学化学中的多尺度问题分析

• 批准号: EP/H05023X/1
• 负责人: Johannes Zimmer
• 金额: $ 2.06万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2010
• 资助国家: 英国
• 起止时间: 2010 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH05023X%2F1
• 关键词: Analysis; multi; scale; problems; mathematical

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.

在过去的几十年里，人们对微米和纳米尺度的科学和工程的兴趣激增，这催生了纳米技术和生物分子建模等全新领域。因此，起源于 20 世纪 50 年代统计力学的一个分支的分子动力学 (MD) 已成为化学、物理和生物学等许多应用科学领域的关键技术，也是材料科学和工程许多应用中的新兴方法。 MD 给数学家带来了令人着迷的挑战，因为系统庞大、复杂、多尺度，尤其是离散的。离散模型的系统研究仍处于起步阶段。巴斯的数学家和科学家团队已经确定了我们想要详细了解的三个问题：（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实现多种配置。从一种配置更改为另一种配置的情况很少见，但通常很重要。我们怎样才能可靠地计算出相应的轨迹呢？ （ii）动态相变：一个令人着迷的新主题是轨迹空间中的相变，这是由玻璃材料引起的。我们的目标是迈出迈向此类时空相变数学理论的第一步。 (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进行建模的。虽然第一个主题旨在理解此类景观中的轨迹，但我们还寻求使用从头计算的数据对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 用户接触最先进的数学技术。我们将这项活动视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 模拟中更强大和更有效的技术，而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开发。例如，DNA 动力学分析对于计算哈密顿轨迹的数学算法来说是一个具有挑战性的测试用例。我们的目标是将巴斯最近开发的方法应用于这样的测试案例，即蛋白质 G (9)，尽管其相对较小（56 个残基），但仍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MD领域取得丰硕成果，MD将成为本世纪的主导技术。

项目成果

Upscaling from particle models to entropic gradient flows

从粒子模型升级到熵梯度流

• DOI: 10.1063/1.4726509
• 发表时间: 2012
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Dirr N

On selection criteria for problems with moving inhomogeneities

运动不均匀性问题的选择标准

• DOI: 10.48550/arxiv.1011.4596
• 发表时间: 2010
• 作者: Herrmann M

Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

小旋节区双稳态晶格模型中的亚音速相变波

• DOI: 10.1137/120877878
• 发表时间: 2013
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Herrmann M