1 Introduction

The virial stress is commonly used to find the macroscopic (continuum) stress in molecular dynamics computations. The macroscopic stress tensor in a macroscopically small, but microscopically large, volume $\Omega$ is typically taken to be:

$$\sigma_{\alpha\beta} = \frac1\Omega \sum_{i {\rm\ in\ } \Omega} \... ...e{\frac {1}{2}}} \sum_j (x^{(j)}_\alpha - x^{(i)}_\alpha) f^{(ij)}_\beta\right)$$ (1)

where $m^{(i)}$ is the mass of the $i$ -th molecule in $\Omega$ , $\vec x^{(i)}$ its position, with Cartesian components $(x_1^{(i)},x_2^{(i)},x_3^{(i)})=(x^{(i)},y^{(i)},z^{(i)})$ , $\vec u^{(i)}$ its velocity, $\bar{\vec u}$ the local average velocity, and $\vec f^{(ij)}$ is the force on molecule $i$ exerted by another molecule $j$ .

This stress was found by Irving & Kirkwood (1950) for the ensemble-averaged equations of hydrodynamics, though they did not write down the local volume-averaged version above. The version that will be adopted here was derived by Cormier et al. (2001), based on the work of Lutsko (1988). Similar expressions have been derived by earlier authors, for example based on the virial of Clausius (1870).

Recently Zhou (2003$a$ ,$b$ ) has cast doubt on the validity of the first, dynamical, term in the virial formula since the desired Cauchy stress is supposed to represent mechanical forces only. In this paper we will examine the proper stress from the most basic ideas, in order to determine the physical meaning of both terms unambiguously.

Figure 1: Straightforward evaluation of the macroscopic stress.

We will be concentrating on the molecular dynamics simulation of a solid. We assume the solid is at rest on the macroscopic scale, so that the average velocity vanishes, and that the solid is homogeneous on all but the microscopic level.

Figure 1 shows a direct attempt to evaluate the stress on a surface $AA'$ by finding the force exerted on the set $I$ of atoms below $AA'$ by the set $J$ above $AA'$ . For convenience, we have taken the $y$ -axis normal to the surface $AA'$ , which extends distances $\Delta x$ and $\Delta z$ in the other two directions.

We assume that while the surface is macroscopically small, it extends over a large enough microscopic area that microscopic variations are averaged away; (compare Nakane 2000). In addition we assume it is large enough compared to the molecular interaction distance (sketched as grey) that edge effects on the computed net force are negligible. By its very definition, our macroscopic stress becomes:

$$\vec \sigma_2 = \frac1{\Delta x\Delta z} \sum_{i {\rm\ in\ } I} \sum_{j {\rm\ in\ } J} \vec f^{(ij)}$$ (2)

Of course, to reduce random fluctuations, in addition to averaging over the spatial intervals $\Delta x$ and $\Delta z$ , one might want to average further over a macroscopically small time interval around the desired time of the stress. However, a purely spatial average tends to be more convenient in a time-marching computation, and one expects the ergodic assumption to be valid that averaging short scale processes over time can be simulated by averaging different stages of those processes over space.

In section 3, it will be seen that the sum of forces (1.2) is directly evaluated using the second term in the virial sum (1.1), begging the question what the first term is doing. Certainly, as Zhou (2003$a$ ,$b$ ) very correctly explains, if the sets $I$ and $J$ are Lagrangian sets, (i.e. they contain the same atoms at different times,) the sum of forces (1.2) is all there is; absolutely no additional dynamical terms should be added.