Generalised Fluctuating Dynamic Density-Functional Theory: Derivation and Applications

Abstract

Classical density-functional theory is a powerful theoretical framework for studying the equilibrium density distribution of complex many-body systems. The core of the framework is to bridge microscopic and macroscopic descriptions of fluids through a variational principle that minimises the system's free energy. For dynamical settings, dynamic DFT (DDFT) extends the equilibrium framework to capture the time evolution of an order parameter, typically density. However, DDFT is often insufficient for mesoscopic systems far from equilibrium where fluctuations become significant, necessitating a stochastic extension: fluctuating DDFT. Here we derive a generalised FDDFT from first principles starting from the microscopic Hamiltonian dynamics using coarse-graining techniques, including Zwanzig's projection operator method. The resulting FDDFT applies to arbitrary observable fields, where the dynamics are driven by a thermodynamic force expressed in terms of an entropy functional. The dynamics consist of reversible, dissipative, and stochastic contributions, each directly linked to the underlying microscopic dynamics via the Liouville operator, Green-Kubo relations, and the fluctuation-dissipation theorem. To highlight the difference in the level of description between DDFT and FDDFT, we demonstrate how further coarse-graining via averaging leads to a generalised DDFT, valid at the macroscale. To showcase the effectiveness and utility of the generalised FDDFT, we apply it to two prototypical fluid systems: colloidal and simple viscous fluids. In the former case, we utilise the new formalism to re-derive the recently found colloidal FDDFT. In the latter case, we rigorously recover the non-isothermal extension of the Landau-Lifshitz fluctuating Navier-Stokes equations, providing fundamental derivation of this well-known phenomenological model, enabling to clarify its underlying principles.