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

用统计力学方法描述各类经典胶体流体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。这需要适当的理论，预测，准确，并在数字上实现。经典的动态密度泛函理论（DDFT）是一个很有前途的选择，然而，它面临着基于有时不受控制的近似的挑战。因此，从严格的数学角度对DDFT进行基本理解，并探索与最近提出的理论改进的关系，是非常重要的。本研究项目的目的是在SPP 2265的第一个资助期内继续我们的工作，并阐明DDFT的数学背景和相关方法。我们将继续与spp2265其他成员开展成功的合作。在SPP 2265的第二个资助期，我们的项目围绕三个支柱发展。(I)开发、改进和应用随机外部影响下流体的近似DDFT模型，包括活度、细菌生长、奇数扩散系统和随机多孔介质。（二）平衡点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，如位置相关的活动速度，还要考虑两体水平上的时间相关的密度势映射。（三）DDFT成分或结果的数学评价。这一点涉及到对一类自由能泛函的严格分析，它进入许多DDFT方法和平衡极限，以及从DDFT角度解决超均匀性的新概念。

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.