We present a finite element discretization of a non-linear diffusion equation used in the field of critical phenomena and, more recently, in the context of Dynamic Density Functional Theory. The discretized equation preserves the structure of the continuum equation. Specifically, it conserves the total number of particles and fulfills an H-theorem as the original partial differential equation. The discretization proposed suggests a particular definition of the discrete hydrodynamic variables in microscopic terms. These variables are then used to obtain, with the theory of coarse-graining, their dynamic equations for both averages and fluctuations. The hydrodynamic variables defined in this way lead to microscopically derived hydrodynamic equations that have a natural interpretation in terms of discretization of continuum equations. Also, the theory of coarse-graining allows to discuss the introduction of thermal fluctuations in a physically sensible way. The methodology proposed for the introduction of thermal fluctuations in finite element methods is general and valid for both regular and irregular grids in arbitrary dimensions. We focus here on simulations of the Ginzburg-Landau free energy functional using both regular and irregular 1D grids. Convergence of the numerical results is obtained for the static and dynamic structure factors as the resolution of the grid is increased.

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We study the process of coarse-graining in a simple model of diffusion of Brownian particles. At a detailed level of description, the system is governed by a Brownian dynamics of non-interacting particles. The coarse-level is described by discrete concentration variables defined in terms of Delaunay cells. These coarse variables obey a stochastic differential equation that can be understood as a discrete version of a diffusion equation. We study different models for the two basic building blocks of this equation which are the free energy function and the diffusion matrix. The free energy function is shown to be non-additive due to the overlapping of cells in the Delaunay construction. The diffusion matrix is state dependent in principle, but for near-equilibrium situations it is shown that it may be safely evaluated at the equilibrium value of the concentration field.

Density Functional Theory (DFT) is a successful and well-established theory for the study of the structure of simple and complex fluids at equilibrium. The theory has been generalized to dynamical situations when the underlying dynamics is diffusive as in, for example, colloidal systems. However, there is no such a clear foundation for Dynamic DFT (DDFT) for the case of simple fluids in contact with solid walls. In this work, we derive DDFT for simple fluids by including not only the mass density field but also the momentum density field of the fluid. The standard projection operator method based on the Kawasaki-Gunton operator is used for deriving the equations for the average value of these fields. The solid is described as featureless under the assumption that all the internal degrees of freedom of the solid relax much faster than those of the fluid (solid elasticity is irrelevant). The fluid moves according to a set of non-local hydrodynamic equations that include explicitly the forces due to the solid. These forces are of two types, reversible forces emerging from the free energy density functional, and accounting for impenetrability of the solid, and irreversible forces that involve the velocity of both the fluid and the solid. These forces are localized in the vicinity of the solid surface. The resulting hydrodynamic equations should allow one to study dynamical regimes of simple fluids in contact with solid objects in isothermal situations.

The theory of nonlocal isothermal hydrodynamics near a solid object derived microscopically in the study by Camargo et al. [J. Chem. Phys. 148, 064107 (2018)] is considered under the conditions that the flow fields are of macroscopic character. We show that in the limit of macroscopic flows, a simple pillbox argument implies that the reversible and irreversible forces that the solid exerts on the fluid can be represented in terms of boundary conditions. In this way, boundary conditions are derived from the underlying microscopic dynamics of the fluid-solid system. These boundary conditions are the impenetrability condition and the Navier slip boundary condition. The Green-Kubo transport coefficients associated with the irreversible forces that the solid exert on the fluid appear naturally in the slip length. The microscopic expression for the slip length thus obtained is shown to coincide with the one provided originally by Bocquet and Barrat [Phys. Rev. E 49, 3079 (1994)].