Virial stress

Virial stress is a measure of mechanical stress on an atomic scale for homogeneous systems.
The expression of the (local) virial stress can be derived as the functional derivative of the free energy of a molecular system with respect to the deformation tensor.
Volume averaged Definition.
The instantaneous volume averaged virial stress is given by
where 
At zero kelvin, all velocities are zero so we have
This can be thought of as follows. The τ11 component of stress is the force in the "x"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.
The volume averaged virial stress is then the ensemble average of the instantaneous volume averaged virial stress.
In a three dimensional, isotropic system, at equilibrium the "instantaneous" atomic pressure is usually defined as the average over the diagonals of the negative stress tensor:
The pressure then is the ensemble average of the instantaneous pressure
This pressure is the average pressure in the volume formula_4.
Equivalent Definition.
It's worth noting that some articles and textbook use a slightly different but equivalent version of the equation
where formula_17 is the "i"th component of the vector oriented from the formula_3th atoms to the "k"th calculated via the difference
formula_19
Both equation being strictly equivalent, the definition of the vector can still lead to confusion.
Derivation.
The virial pressure can be derived, using the virial theorem and splitting forces between particles and the container or, alternatively, via direct application of the defining equation formula_20 and using scaled coordinates in the calculation.
Inhomogeneous Systems.
If the system is not homogeneous in a given volume the above (volume averaged) pressure is not a good measure for the pressure. In inhomogeneous systems the pressure depends on the position and orientation of the surface on which the pressure acts. Therefore, in inhomogeneous systems a definition of a local pressure is needed. As a general example for a system with inhomogeneous pressure you can think of the pressure in the atmosphere of the earth which varies with height.
Instantaneous local virial stress.
The (local) instantaneous virial stress is given by:
formula_21
Measuring the virial pressure in molecular simulations.
The virial pressure can be measured via the formulas above or using volume rescaling trial moves.