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

A Russo and MA Duran-Olivencia and P Yatsyshin and S Kalliadasis, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 53, 445007 (2020).

DOI: 10.1088/1751-8121/ab9e8d

This work introduces a theoretical framework to describe the dynamics of reacting multi-species fluid systems in-and-out of equilibrium. Our starting point is the system of generalised Langevin equations which describes the evolution of the positions and momenta of the constituent particles. One particular difficulty that this system of generalised Langevin equations exhibits is the presence of a history-dependent (i.e. non-Markovian) term, which in turn makes the system's dynamics dependent on its own past history. With the appropriate definitions of the local number density and momentum fields, we are able to derive a non- Markovian Navier-Stokes-like system of equations constituting a generalisation of the Dean-Kawasaki model. These equations, however, still depend on the full set of particles phase-space coordinates. To remove this dependence on the microscopic level without washing out the fluctuation effects characteristic of a mesoscopic description, we need to carefully ensemble-average our generalised Dean-Kawasaki equations. The outcome of such a treatment is a set of non-Markovian fluctuating hydrodynamic equations governing the time evolution of the mesoscopic density and momentum fields. Moreover, with the introduction of an energy functional which recovers the one used in classical density- functional theory and its dynamic extension (DDFT) under the local- equilibrium approximation, we derive a novel non-Markovian fluctuating DDFT (FDDFT) for reacting multi-species fluid systems. With the aim of reducing the fluctuating dynamics to a single equation for the density field, in the spirit of classical DDFT, we make use of a deconvolution operator which makes it possible to obtain the overdamped version of the non-Markovian FDDFT. A finite-volume discretization of the derived non- Markovian FDDFT is then proposed. With this, we validate our theoretical framework in-and-out-of-equilibrium by comparing results against atomistic simulations. 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.

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