A Finite-Volume Method for Fluctuating Dynamical Density Functional Theory

TL;DR

This paper presents a finite-volume numerical scheme for stochastic gradient-flow equations in fluctuating density functional theory, enabling accurate simulations of phenomena involving thermal fluctuations and energy barriers.

Contribution

The paper introduces a novel positivity-preserving finite-volume scheme for stochastic DFT equations that handles general free-energy functionals and captures energy barrier crossing dynamics.

Findings

Accurately reproduces statistical properties like structure factors and correlations.

Simulates energy barrier crossing dynamics beyond mean-field approaches.

Handles general free-energy functionals including external fields and interactions.

Abstract

We introduce a finite-volume numerical scheme for solving stochastic gradient-flow equations. Such equations are of crucial importance within the framework of fluctuating hydrodynamics and dynamic density functional theory. Our proposed scheme deals with general free-energy functionals, including, for instance, external fields or interaction potentials. This allows us to simulate a range of physical phenomena where thermal fluctuations play a crucial role, such as nucleation and other energy-barrier crossing transitions. A positivity-preserving algorithm for the density is derived based on a hybrid space discretization of the deterministic and the stochastic terms and different implicit and explicit time integrators. We show through numerous applications that not only our scheme is able to accurately reproduce the statistical properties (structure factor and correlations) of the physical system, but, because of the multiplicative noise, it allows us to simulate energy barrier crossing dynamics, which cannot be captured by mean-field approaches.