Abstract
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.
Highlights
Many applications, from chemistry to biology and engineering, exploit complex fluids consisting of colloidal particles
A rigorous and systematic derivation of fluctuating hydrodynamics (FH) and fluctuating dynamical DFT (DDFT) (FDDFT) for the general case of arbitrarily shaped thermalized particles was offered by Durán-Olivencia et al [34]
Equilibrium mono-component system In order to test the numerical stochastic integrator, we consider a non-Markovian FDDFT approach to model a system of N non-reacting ideal gas particles with single exponential memory kernel
Summary
From chemistry to biology and engineering, exploit complex fluids consisting of colloidal particles. A rigorous and systematic derivation of FH and fluctuating DDFT (FDDFT) for the general case of arbitrarily shaped thermalized particles was offered by Durán-Olivencia et al [34] It should be emphasised, that the DK model describes the evolution of instantaneous microscopic fields, containing the same physical information as the original set of GLEs, and leaves the equations computationally intractable and at the same time disconnected from the original Landau et al theory. Some systems are characterized by a time-scale separation between resolved and unresolved particle dynamics, due for instance to the much higher inertia of the first compared to the latter particles In these cases, the memory kernel tensor in equation (1) can be approximated as θs(t) = 2θs,0δ(t), where δ(t) is a Dirac’s delta function and θs,0 is a constant. Applying the operator I in equation (29) for a multicomponent system of ideal gases, we obtain the following non-Markovian time evolution of the density field of the species s:. For memory kernels in the form of exponential series, we simplified the convolutional structure of the above equation, defining an effective diffusion coefficient including low-frequencies contributions to memory effects
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