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

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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.