Fluctuations in density-functional theory for fluids: Theory and computations

Classical density-functional theory (DFT) for fluids and its dynamic extension (DDFT) provide an appealing mean-field framework for describing equilibrium and dynamics of complex soft matter systems. For a long time, accounting for the effects of thermal fluctuations under this theoretical framework was thought to be impossible, if not fundamentally incorrect. Here, we present an ab-initio derivation of a fluctuating DDFT (FDDFT) where thermal fluctuations are derived from first principles, thus advancing the long-standing debate about the inclusion of fluctuations. As a by-product, we obtain a non-equilibrium energy functional which recovers the Helmholtz free energy under equilibrium conditions. We show that there is a one-to-one connection between our FDDFT formalism and the phenomenological Landau-Lifshitz fluctuating hydrodynamics. To showcase the potential and computational capacity of the formalism developed in this work, we apply it to the study of homogeneous nucleation, a classical problem considered to be outside the limits of applicability of DFT. Finally, we highlight how our theoretical effort can open the door to the discovery of new physical laws governing the dynamics of soft-matter systems out-of-equilibrium and under noise-dominated conditions.