The overdamped limit of dynamic density functional theory: rigorous results (Q2913941)

The authors present a Smoluchowski-type evolution equation that is derived for the one-particle distribution function for two-body hydrodynamic interactions and one- and two-body potentials. This new equation includes a novel definition of the diffusion tensor.NEWLINENEWLINEThe paper is divided into five sections and one appendix. After the Introduction with an extended review of dynamic density functional theory (DDFT), in Section 2, the authors give (i) a description of the model, in both the original and rescaled timescales, (ii) assumptions, and (iii) an overview of the main result. In Section 3, a solvability condition for the Hilbert expansion of the one-body distribution \(f^{(1)}\) is developed, which forms the basis for the proof of the main result stated in Section 4. This section also presents the relationship with existing formulations of the one-body Smoluchowski equation. In Section 5, the impact of the main result, including its application to the derivation of the DDFT, is discussed. Appendix A contains proofs of lemmas from Section 3.