Macroscopic equations for microscopic transport processes

通常，工程材料由大约10^23个原子或分子组成，材料的行为是由其所有粒子的相互作用决定的。尽管有大量的单个粒子，但各种各样的流动和传热问题可以用相当少的偏微分方程来描述，例如众所周知的流体和气体流动的纳维-斯托克斯方程和傅立叶方程，或者菲克扩散定律。由于粒子之间的强烈相互作用（例如气体分子的碰撞），干扰有效地分布在粒子之间，因此在许多应用中，粒子的行为不是作为个体，而是作为一个连续体。然而，连续体的响应来源于微观尺度上的过程，这为研究微观和宏观行为之间的关系提供了重要的理由。先进材料的宏观模型——例如：用于燃料电池的PEM膜需要许多传输系数，而对微观行为的理解可以为后者的建模提供信息。在更极端的条件下——例如：稀薄的气体流动、微流动、非平衡蒸发和冷凝以及其他界面过程——标准的工程定律失效了，必须被先进的模型所取代，这些模型可以在原子尺度上解释这些过程的更多细节。现代工程设计依赖于器件及其过程的数值模拟。良好而准确的数学模型对于这项任务是必不可少的。提出的研究考虑了微观和宏观世界之间的关系，以推导和验证稀薄气体、用于燃料电池的聚合物膜以及蒸发和冷凝过程的数学传输模型。由此产生的模型将在宏观层面上描述过程，即作为一个连续体，但包含微观物质行为的知识。因此，它们将超越标准的工程定律，为这些复杂而重要的过程的高级模拟打开大门。

Typically, engineering materials consist of about 10^23 atoms or molecules, and the behavior of a material results from the interplay of all its particles.  Despite this huge number of individual particles, a wide variety of flow and heat transfer problems can be described by a rather low number of partial differential equations, e.g. the well-known equations of Navier-Stokes and Fourier for fluid and gas flows, or Fick's law of diffusion. Due to the intense interaction between the particles (e.g. collisions of gas molecules), disturbances are effectively distributed between the particles, so that in many applications the particles behave not as individuals, but as a continuum.      Nevertheless, the response of the continuum results from processes on the microscopic scale, and this gives an important reason to study the relation between the microscopic and the macroscopic behavior.  Macroscopic models for advanced materials--e.g. PEM membranes for fuel cells--require many transport coefficients, and understanding of the microscopic behavior can inform the modelling of the latter.      Under more extreme conditions--e.g. rarefied gas flows, microflows, non-equilibrium evaporation and condensation and other interface processes--the standard engineering laws break down, and must be replaced by advanced models, which account for more details of the processes on the atomic scale.      Modern engineering design relies on numerical simulations of devices, and the processes therein. Good and accurate mathematical models are indispensable for this task. The proposed research considers the relation between the micro- and the macroworld in order to derive, and validate, mathematical transport models for rarefied gases, polymer membranes for use in fuel cells, and evaporation and condensation processes. The resulting models will describe the processes on a macroscopic level, i.e. as a continuum, but incorporate knowledge from the microscopic material behavior. Thus, they will go beyond the standard engineering laws and open the door for advanced simulations of these complicated and important processes.