Field-Theoretic Simulation of Dean-Kawasaki Dynamics for Interacting Particles

TL;DR

This work tackles the challenge of connecting microscopic particle dynamics to a mesoscopic field description for interacting liquids via Dean–Kawasaki mappings. It implements a mathematically regularized, coarse-grained RIDK framework and a grid-based GRID CG scheme to enable stable, area- and time-resolved simulations of fluctuating hydrodynamics. The authors demonstrate that coarse-graining with a kernel width $\\varepsilon$ and grid spacing $h$ can be reconciled through the empirical relation $h \,\\approx\, 2\\varepsilon$, and they provide scaling analyses and error bounds to guide parameter choices. Through Gaussian-core and Lennard-Jones model studies, RIDK with GRID CG reproduces essential structural correlations of the microscopic reference, showing promise for efficient multiscale simulations of liquids, polymers, and glasses.

Abstract

The formulation of a fluctuating hydrodynamic theory for interacting particles is a crucial step in the theoretical description of liquids. The microscopic mappings proposed decades ago by Dean and Kawasaki have played a central role in the analytical treatment of such problems. However, the singular mathematical nature of the density distributions used in these derivations raises concerns about the validity and practical utility of the resulting stochastic partial differential equations, particularly for direct numerical simulations. Recent efforts have centered on establishing a rigorous coarse-graining procedure to regularize the effective Dean-Kawasaki equation. Building on this foundation, we numerically investigate weakly interacting fluids within such a regularized framework for the first time. Our work reveals, at the level of structural correlations, the effects of regularization on the Dean-Kawasaki formalism and paves the way for improved numerical approaches to simulate fluctuating hydrodynamics in liquids.

Figures (10)

• Figure 1: Schematic diagram illustrating the path taken in this work: bridging microscopic particle dynamics and mesoscopic field dynamics via Dean's equation. Since the original Dean's equation, involving microscopic density, interactions, and correlations (red), cannot be directly applied at the mesoscopic field level, mesoscopic coarse-graining is required under the RIDK framework (blue). From the mathematically regularized fields $(\rho_\varepsilon, j_\varepsilon)$, we perform a scaling analysis to approximate these fields as $(\rho,j)$ under specific conditions, allowing us to efficiently propagate the field dynamics of $(\rho,j)$ using the finite element method.
• Figure 2: Effect of mesoscopic coarse-graining on molecular configuration. Here, we compare various levels of mesoscopic coarse-graining applied to the MD simulation snapshot of the Gaussian core model (box size: $L=2\pi$) by imposing Gaussian kernels with variances $\varepsilon^2$ from $\varepsilon=L/100$ to $L/20$ at each particle position. For small $\varepsilon$ (left), the coarse-grained configuration resembles the microscopic trajectory, while larger $\varepsilon$ (right) smooths out microscopic details, consistent with $\rho_\varepsilon(\mathbf{k},t)=e^{-\tfrac{1}{2}\mathbf{k}^2\varepsilon^2}\hat{\rho}(\mathbf{k},t)$.
• Figure 3: Structural correspondence between the number of numerical grid points $n_g$ (or grid size $h$) used in the finite element method and the mesoscopic coarse-graining length $\varepsilon$ for interacting particles. This correspondence was established by examining the coarse-grained RDF, equivalent to $g(r)$ defined as $g(\textbf{r}):=\langle \sum_i \delta(\mathbf{r}-\mathbf{r}_i)\rangle/\rho$, at varying $n_g$ and $\varepsilon$ values. We find that two differently coarse-grained RDFs are nearly identical when $h=2\varepsilon$ across different coarse-graining levels: (a, f) $\varepsilon=0.06\,\textrm{\AA}$ with $n_g=250$, (b, g) $\varepsilon=0.15\,\textrm{\AA}$ with $n_g=100$, (c, h) $\varepsilon=0.3\,\textrm{\AA}$ with $n_g=50$, (d, i) $\varepsilon=0.6\,\textrm{\AA}$ with $n_g=25$, and (e, j) $\varepsilon=0.75\,\textrm{\AA}$ with $n_g=20$. Notably, this finding also holds for different interaction types: (a-e) Gaussian core model and (f-j) bare Lennard-Jones model, where the RDFs are indistinguishable from those of the regularized potentials (see Sec. IX)
• Figure 4: Essential steps underlying the GRID CG method for determining mesoscopic coarse-grained energetics. (Left): Initially prepared microscopic MD trajectory of the system of interest due to the bottom-up nature of our approach. (Middle): Construction of an approximated mesoscopic density histogram at the specified coarse-grained resolution ($\varepsilon$), where the grid size $h$ corresponds to $2\varepsilon$ based on the established correspondence. (Right): Determination of mesoscopic interaction $V_\mathrm{GRID}$ between grid elements by matching it to the microscopic energetics between particles within each grid element using Eq. (\ref{['eq:ECG']}).
• Figure 5: Structural correlations (RDF) of the Gaussian core model at different scales: (a) microscopic and (b) mesoscopic. In panel (b), using the microscopic RDF (gray) with a weakly structured profile, the mesoscopic coarse-grained RDFs are shown for various grid sizes $n_g$: 5 (red), 10 (orange), 20 (green), 25 (blue), 50 (purple), and 80 (navy). As $n_g$ decreases (or $\varepsilon$ increases), the structural correlations gradually diminish, consistent with Fig. \ref{['fig:figure4']}.