Virial stress

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Virial stress is a measure of mechanical stress on an atomic scale. It is given by

$\tau_{ij} = \frac{1}{\Omega} \sum_{k \in \Omega} \left(-m^{(k)} (u_i^{(k)}- \bar{u}_i) (u_j^{(k)}- \bar{u}_j) + \frac{1}{2} \sum_{\ell \in \Omega} ( x_i^{(\ell)} - x_i^{(k)}) f_j^{(k\ell)}\right)$

where

$k$ and $\ell$ are atoms in the volume,
$Ω$ ,
$m (k)$ is the mass of atom k,
$u_i^{(k)}$ is the i^th component of the velocity of atom k,
$\bar{u}_j$ is the j^th component of the average velocity of atoms in the volume,
$x_i^{(k)}$ is the i^th component of the position of atom k, and
$f_i^{(k\ell)}$ is the i^th component of the force between atom $k$ and $\ell$ .

At zero kelvin, all velocities are zero so we have

$\tau_{ij} = \frac{1}{2\Omega} \sum_{k,\ell \in \Omega} ( x_i^{(\ell)} - x_i^{(k)}) f_j^{(k\ell)}$ .

This can be thought of as follows. The τ₁₁ component of stress is the force in the 1 direction divided by the area of a plane perpendicular to that direction. Consider two adjacent volumes separated by such a plane. The 11-component of stress on that interface is the sum of all pairwise forces between atoms on the two sides....