Exploring the boundary between atoms and the continuum by computers: a personal history

Abstract

In this admittedly personal account of the history of atomistic simulations of fluids (at the atomic or molecular level), I will focus on the competing efforts to reach the boundary between atoms and the continuum. The prevailing wisdom was that thermal fluctuations at the atomistic scale—both time (a few mean collision times) and space (a few atomic spacings)—would make the connection virtually impossible. This is just a part of the story about how molecular dynamics was able to connect to Navier–Stokes–Fourier hydrodynamics. Resistance in the theoretical physics community to computer simulations of equilibrium fluids at the atomistic scale was only exceeded by the even stiffer objections to non-equilibrium molecular-dynamics simulations: after the fifty years from Boltzmann to molecular dynamics, it took another quarter century to overcome the doubts.

Appendix: Correcting self-diffusion by the long-time tail

To obtain the self-diffusion coefficient D in d dimensions, you need to integrate the velocity autocorrelation function (vacf) from zero to infinity in time. Out to a couple of mean collision times, the decay of the vacf is rapid (approximately exponential for hard disks in 2D or hard spheres in 3D), and its time-integral gives what is referred to as the Enskog-theory value, $D_{E}$ Beyond that early time of a few collisions, the hydrodynamic long-time tail¹⁸ of the vacf goes like $t^{-d/2}$, until the leading sound wave from the double vortex (moving at c, the adiabatic bulk sound speed) reaches the edge of the periodic computational box of volume $V = L^{d} \sim N$, at the sound-traversal time $t_{L} = L/c$. Thereafter, the double vortex quickly decays exponentially, due to interaction with its periodic images, and the hydrodynamic correction to D in a finite system thus depends on $t_{\mathrm {L}} \sim N^{1/d}$.

For hard spheres in 3D, the time-integral of $t^{-3/2}$ approaches $-t_{L}^{-1/2} \sim -(N^{1/3})^{-1/2} = -N^{-1/6}$. Thus, $D =D_{\infty } - {\textit{const.}}$ $N^{-1/6}$, which displays a finite hydrodynamic limit in 3D, as $N \rightarrow \infty$.

On the other hand, for hard disks in 2D, the time-integral of $t^{-1}$ approaches $\hbox {ln}(t_{L})\sim \hbox {ln}(N^{1/2}) \sim \hbox {ln}(N)$. Thus, $D = D_{E} + {\textit{const.}}$ ln(N), which shows that there is no bound (i.e., no hydrodynamic limit) in 2D, as $N\rightarrow \infty$.

In 2D, Holian’s Maxwell Demon NEMD results for $N = 168$, 1512, and 5822 hard disks (Erpenbeck and Wood 1977), can be fitted over the entire range of system sizes by nonlinear least squares,

$$\begin{aligned} D/D_{E} = 0.757 + 0.126 ~\hbox {ln}(N) , \end{aligned}$$

while in 3D, the Demon’s results for $N = 108$, 500, and 4000 hard spheres (Erpenbeck and Wood 1977) can be fitted by nonlinear least-squares,

$$\begin{aligned} D/D_{E} = 1.44 - 0.580 \, N^{-1/6} . \end{aligned}$$

Alder et al.’s Green–Kubo equilibrium MD results for $N = 108$ and 500 hard spheres (Alder et al. 1970a), can be fitted by $D/D_{E} = 1.43 - 0.585\,N^{-1/6}$, which agrees very well indeed with NEMD.