4 Final remarks

Most of the examples that Zhou (2003$a$ ) uses to argue the incorrectness of the virial stress suffer from the problem that the macroscopic motion is not substracted out. We have absolutely no issue with the claim that the macroscopic motion part does not corresponds to physical forces. This is, after all, immediately clear from doing a Gallilean transform. Nor does it seem that there has been much misconception about this; it is very clearly shown in Irving & Kirkwood (1950), and authors that needed to substract the mean motion seem to have done so, eg Hardy (1982).

(Yet, as a fluid dynamicist, the author wants to point out that the Lagrangian description, following the motion of the fluid, is most times highly inconvenient for us. In a fixed coordinate frame, the additional terms in the virial stress become simply the convective terms that must be added to the Eulerian conservative equations. The author is, however, unaware of claims that the convective terms would be physical forces.)

Figure 11 in Zhou (2003$a$ ) does include the thermal motion, but it is assumed to be negligible. Only the example of figure 5 actually addresses a valid case of thermal motion effects. It is no more than a symmetric two-atom system were the atoms $i$ and $j$ , connected by a linear or nonlinear spring, vibrate along the $y$ -axis (his $x$ -axis) according to $y^{(j)}=-y^{(i)}$ . One can fill space with such pairs to create a macroscopic continuum. Obviously, this does not support any macroscopic force.

Zhou's error in this example seems to be simple sloppy math. He finds the macroscopic virial stress by time-averaging over a period $\tau$ , which is appropriate for his assumption that all pairs are in the same phase, (the assumption of random phases would allow a more appealing space average, and would not change the results here.) His error, in his equation (31), is in assuming that the average of the second term in the virial stress is zero. In fact, the average of $\left(y^{(j)}-y^{(i)}\right) f_y^{(ij)}$ is not just the average of $\left(y^{(j)}-y^{(i)}\right)$ times the average of $f_y^{(ij)}$ , which would be zero, but also includes the average of $\left({y'}^{(j)}-{y'}^{(i)}\right) {f'}_y^{(ij)}$ , where primes indicate instantaneous deviation from the average. The last term is always positive: when the atoms are closer together than nominal the force between them will be repulsive and when they are farther apart, attractive. The correct average of the second term in the virial stress is in fact easily found as

$$\frac 1{\tau\Omega} \int_\tau - 2 y^{(i)} f_y^{(ij)} \/{\rm d}t =... ...^{(i)} \/{\rm d}t = \frac 1{\tau\Omega} \int_\tau 2 m \dot y^{(i)2} \/{\rm d}t,$$ (3)

using integration by parts and periodicity. This, of course, is just what is needed to balance the first, dynamical term, and the stress is correctly evaluated to be zero as it should, and is not a negative value as Zhou claims.

This author feels that it is also misleading to claim that for a gas, the dynamical term in the virial stress corresponds to a physical force at the wall only. While this, obviously, is true for the physics-book model of an ideal gas as a system of noninteracting point masses, a system at such a Knudsen number does not truly act as a simple continuum on macroscopically small scales. The gasses that ordinary fluid mechanics deals with have a free path length that is small compared to macroscopic scales. We propose that the discussion of section 2 can then be changed to read that atom $i$ wanders into the $J$ region to lose its momentum in collisions with the atoms there and becomes anonymous within its new surroundings at a distance that scales with the free path length. To maintain the statistical mass distribution, an other atom will come wandering out of that region to cross back into region $i$ . One might, with some justification, object that due to the atom swap, the region $I$ is no longer representative of any Lagrangian set. We submit however that Newton's laws for systems are not at all affected if we swap two identical particles with the same momentum. However one might think about that, it is obviously unjustified to say that the collision forces that make atom $i$ anonymous in set $J$ and the ones that kick the new atom back to set $I$ are not internal mechanical forces ``in any sense.'' Assuming that the atoms only fill a small fraction of space, so that collisions with the atoms at opposite sides of the plane $AA'$ are rare, most of those physical forces will be represented by the dynamical term.

A note on the virial averaging might also be in order. The straight Cormier et al. average of the Kirkwood & Irving stress has the disadvantage that it also affects the macroscopic scales to second order in averaging size. Thus it seems that it may in many cases be more desirable to use the higher order operators commonly used in discrete vortex methods (eg, Shankar & Van Dommelen 1996). The advantage would be that for given error in smoothing macroscopic scales, one can average over a larger region in space, reducing microscopic variations. This seems to be equivalent to the expressions derived by Hardy (1982), though higher order averaging functions are not uniformly positive. In any case, now that we understand the relevance of the virial stress, we propose to concentrate on how to evaluate it best.