Memory effects in fluctuating dynamic density-functional theory: theory and simulations*

This work introduces a theoretical stochastic framework to describe the dynamics of reacting multi-species fluids in and out of equilibrium. We present an ab-initio derivation of a non-Markovian Navier-Stokes-like system of equations which constitutes a generalisation of the Dean-Kawasaki model. However, such equations still depend on all the microscopic degrees of freedom. To remove this dependence on the microscopic level without washing out the fluctuation effects, which are characteristic of a mesoscopic description, we ensemble-average our generalised Dean-Kawasaki equations. This results in a set of non-Markovian fluctuating hydrodynamic equations governing the time evolution of the mesoscopic density and momentum fields. By introducing an energy functional which recovers the one used in classical density-functional theory (DFT) under the local-equilibrium approximation, we obtain a non-Markovian fluctuating dynamical DFT (FDDFT) for reacting multi-species fluids. To numerically validate our theoretical framework, we carry out a finite-volume discretisation of our non-Markovian FDDFT. Finally, we illustrate the influence of non-Markovian effects on the dynamics of non-linear chemically reacting fluid systems with a detailed study of memory-driven Turing patterns.