Statistical mechanics of soft matter: Derivation, analysis and implementation of dynamic density functional theories

“软物质”一词用来描述在室温下容易在重力或压力等外力作用下变形的材料，而它们的性质受缓慢的内部动力学控制。软物质通常在工程和生物医学科学中起着核心作用，并具有许多实际应用。提出的研究重点是一大类软物质系统，即经典流体系统，即保持一定体积的粒子系统，并且处于足够高的温度，量子效应可以忽略不计。特别令人感兴趣的是胶体流体，它的颗粒只有微米大小，悬浮在由更多更小更轻的颗粒组成的溶液中，这些颗粒不能用连续统宏观形式（如纳维-斯托克斯方程）来描述。对整个胶体流体的动力学建模是非常昂贵的，而且由于必须考虑大量的粒子和广泛的长度和时间尺度，实际上在计算上是难以处理的。因此，必须采用统计力学方法，最广泛使用的方法是动态密度泛函理论（DDFT）。然而，以前的ddft往往涉及近似值，其准确性和有效性不能先验地确定。因此，从这种公式得到的结果是有问题的，而且往往是不准确的。该提案寻求资助一项为期三年的综合研究计划，该计划旨在进行双管齐下的新颖理论和数值研究，旨在合理理解和系统地预测胶体系统的复杂物理行为和性质。主要目标是发展一种通用的DDFT形式体系，允许对物理相关系统进行准确、系统和预测建模，在这些系统中，以前理想化研究中所有被忽视的影响现在都脱颖而出。这反过来又将有助于逐步提高大量依赖微粒系统的技术和应用的性能和效率。详细的数值模拟将作为分析工作的补充，以核实所开发模型的效力，并协助开发实际应用的工具包。这项研究将由来自爱丁堡大学数学学院和伦敦帝国理工学院化学工程和数学系的团队进行，他们的技能和优势互补：戈达德（复杂多尺度系统，统计力学，分析和计算），Kalliadasis（多尺度流体动力学，理论和计算）和Pavliotis（随机过程，多尺度分析，统计力学）。

The term "soft matter" is used to describe materials which at room temperature can easily deform under external forces like gravity or pressure while their properties are governed by slow internal dynamics. Soft matter often plays a central role in engineering and biomedical science and has numerous practical applications. The proposed research focuses on a large class of soft matter systems, that of classical fluids, i.e. systems of particles which retain a definite volume and are at sufficiently high temperatures that quantum effects can be neglected. Of particular interest are colloidal fluids whose particles are of micrometer size, suspended in a bath of many more, much smaller and lighter particles, which cannot be described by continuum macroscopic formalisms such as the Navier-Stokes equation.Modelling the dynamics of the full colloidal fluids is prohibitively expensive and in fact computationally intractable due to both the large number of particles and the wide range of length- and time-scales which must be considered. As a consequence, one must employ statistical mechanics approaches, with the most widely used method being dynamic density-functional theory (DDFT). However, previous DDFTs often involve approximations whose accuracy and validity cannot be ascertained a priori. As a consequence, the results obtained from such formulations are questionable and often inaccurate.This proposal seeks funding for a comprehensive three-year research programme into a two-pronged novel theoretical and numerical investigation aimed at rationally understanding and systematically predicting the complex physical behaviour and properties of colloidal systems. The primary aim is the development of a generic DDFT formalism that would allow for the accurate, systematic and predictive modelling of physically relevant systems where all the neglected effects in previous idealised studies now come to the fore. This in turn will allow for step improvements to the performance and efficiency of a host of technologies and applications that rely crucially on particulate systems. The analytical work will be complemented by detailed numerical simulations that will act so as to verify the efficacy of the developed models, as well as aiding the development of a toolkit for practical applications. The research will be undertaken by a team from the School of Mathematics of the University of Edinburgh and the Chemical Engineering and Mathematics Departments at Imperial College London with complementary skills and strengths: Goddard (Complex Multiscale Systems, Statistical Mechanics, Analysis and Computations), Kalliadasis (Multiscale Fluid Dynamics, Theory and Computations) and Pavliotis (Stochastic Processes, Multiscale Analysis, Statistical Mechanics).