Uniqueness theorems and analysis of classical density functional theory in nonequilibrium random geometries

用统计力学的方法来描述胶体流体的各种经典模型的时间演化，已成为近几十年来物理学的一个重要研究领域。这就需要适当的理论，这些理论是预测性的，准确的，也是数字上可实现的。一个有前途的选择是经典的动力学密度泛函理论（DDFT），然而，它面临的挑战是基于有时不受控制的近似。因此，从严格的数学角度对DDFT有一个基本的理解，这也是对最近提出的理论改进的一个探索，这是非常重要的。本研究项目的目的是继续我们在SPP 2265的第一个资助期的工作，并阐明DDFT和相关方法的数学背景。我们将继续与SPP 2265的其他成员进行成功的合作。我们在SPP 2265第二个资助期的项目围绕三个支柱发展。(I)发展、改进和应用DDFT近似模型，用于随机外部影响下的流体，包括活性、细菌生长、奇扩散系统和随机多孔介质。(II)平衡态外严格唯一性定理的证明。相应的工作包不仅要考虑额外的随机单体场，如位置相关的主动速度，而且还要考虑两体水平上的时间相关的密度-势映射。(III)DDFT成分或结果的数学评估。这一点涉及到一类自由能泛函，进入许多DDFT方法和平衡极限，以及，解决从DDFT的角度来看，超均匀性上升的概念进行了严格的分析。

Describing the time evolution of various types of classical models for colloidal fluids by methods of statistical mechanics has become an important research area in physics over the last decades. This requires appropriate theories which are predictive, accurate and also numerically realizable. A promising option is classical dynamical density functional theory (DDFT), which, however, faces the challenge of being based on a sometimes uncontrolled approximation. Thus a basic understanding of DDFT from a rigorous mathematical point of view, which also explores the relation to recently proposed improvements of the theory, is of high importance. The aim of this research project is to continue our work in the first funding period of the SPP 2265 and elucidate the mathematical background of DDFT and related approaches. We shall pursue further our successful collaborations with other members of the SPP 2265. Our project in the second funding period of the SPP 2265 evolves around three pillars. (I) Development, improvement and application of approximate DDFT models for fluids under random external influences, including activity, bacterial growth, odd diffusive systems and random porous media. (II) Proofs of rigorous uniqueness theorems out of equilibrium. The corresponding work packages shall not only account for additional random one-body fields, such as a position-dependent active velocity, but also a time-dependent density-potential mapping on the two-body level. (III) Mathematical assessment of ingredients or results of DDFT. This point involves a rigorous analysis of a class of free energy functionals, which enters in many DDFT approaches and also the equilibrium limit, as well as, addressing the rising concept of hyperuniformity from a DDFT perspective.