Unifying renormalized and bare viscosity in two-dimensional molecular dynamics simulations

TL;DR

This work introduces a wavenumber-dependent viscosity $\eta_*(k)$ to unify bare and renormalized transport in two-dimensional molecular dynamics. By computing the equilibrium correlation of time-averaged Fourier components of the fine-grained shear stress, $\eta_*(k)$ captures the divergent $\eta_R(L)$ at small $k$ and the finite $\eta_0$ at large $k$, enabling a direct link between mesoscopic fluctuations and macroscopic transport. The authors show that $\eta_R(L) \approx \eta_*(k)$ at $k = 2\pi\sqrt{2}/L$ and that $S_\infty(\mathbf{k})$ yields $\eta_*(k)$ via $S_\infty(\mathbf{k}) = \frac{4 k_x^2 k_y^2}{k^4} \eta_*(k)$, with $S_\infty(\mathbf{k}) \to \frac{4 k_x^2 k_y^2}{k^4} \eta_0$ at large $k$; applying this to MD data provides estimates $\eta_0 \approx 0.28$ and $a_{\rm uv} \approx 2$, offering a practical route to determine bare transport coefficients from microscopic dynamics. This framework advances understanding of fluctuation-induced transport in 2D and sets the stage for extending the approach to other coefficients and higher dimensions.

Abstract

Fluctuating hydrodynamics provides a framework connecting mesoscopic fluctuations with macroscopic transport behavior. To bridge mesoscopic and macroscopic transport from microscopic dynamics, we introduce a wavenumber-dependent viscosity, defined via the equilibrium correlation of time-averaged Fourier components of the fine-grained shear stress field. Two-dimensional molecular dynamics simulations reveal its small-wavenumber divergence characteristic of the renormalized viscosity, while its large-wavenumber behavior determines the bare viscosity, thereby establishing a link between mesoscopic and macroscopic transport based on microscopic dynamics.

Figures (7)

• Figure 1: $\tau$ dependence of $\eta(L, \tau)$ for various values of $L$.
• Figure 2: System size dependence of $\eta_{\rm R}(L)$.
• Figure 3: Shear stress correlation $S(\bm{k},L)$ as a function of $k$. We choose $\bm{k}$ such that $k_x = k_y = 2\pi n/L$ with $n \in \mathbb{Z}$, which corresponds to $\bm{k} = (k/\sqrt{2}, k/\sqrt{2})$.
• Figure 4: Color display of $S(\bm{k},L=128)$.
• Figure 5: Comparison between $\eta_{\rm R}(L)$ and $\eta_*(k=2\pi\sqrt{2}/L)$. From left to right, the data points correspond to $L = 2, 4, 8, 16, 32, 64, 128$. The dotted line, shown as a guide to the eye, corresponds to $\eta_{*}(2\pi\sqrt{2}/L) = \eta_{\rm R}(L)$. The values of $\eta_*(k)$ are estimated from the simulation data $S(\bm{k},L=128)$.