Coarse-grained hydrodynamics from correlation functions

This paper will describe a formalism for using correlation functions between different grid cells as the basis for determining coarse-grained hydrodynamic equations for modeling the behavior of mesoscopic fluid systems. Configurations from a molecular dynamics simulation or other atomistic simulation are projected onto basis functions representing grid cells in a continuum hydrodynamic simulation. Equilibrium correlation functions between different grid cells are evaluated from the molecular simulation and used to determine the evolution operator for the coarse-grained hydrodynamic system. The formalism is demonstrated on a discrete particle simulation of diffusion with a spatially dependent diffusion coefficient. Correlation functions are calculated from the particle simulation and the spatially varying diffusion coefficient is recovered using a fitting procedure.

Figure 1

Autocorrelation functions for concentration fluctuations in the first five simulation cells. Numbers refer to the center location of the cell.

Figure 2

Cross-correlation functions for concentration fluctuations in the first five simulation cells and the next higher cell. Numbers refer to the center location of the first cell.

Figure 3

Comparison of autocorrelation functions calculated using Eq. (8) and those from simulation. Autocorrelation functions for the cells centered at $-4.5$ and $-0.5$ are shown.

Figure 4

Comparison of cross-correlation functions between cells $i$ and $i-1$ calculated using Eq. (8) and those from simulation. Functions for $i$ corresponding to the cells at positions $-3.5$ and $-0.5$ are shown.

Figure 5

Mean-square concentration fluctuations within cells as a function of position for five separate simulations of $10\times {10}^6$ steps.

Figure 6

Fits to the spatially varying diffusion coefficient $\alpha \left(x\right)$ for different length trajectories.

Figure 7

Values of autocorrelation functions after integrating over time.

Figure 8

Fits from five different simulations (dotted lines) compared with the exact expression for $\alpha \left(x\right)$ (solid line).