Virial stress

The instantaneous volume averaged virial stress is given by

${\displaystyle \tau _{ij}={\frac {1}{\Omega }}\sum _{k\in \Omega }\left(-m^{(k)}(u_{i}^{(k)}-{\bar {u}}_{i})(u_{j}^{(k)}-{\bar {u}}_{j})+{\frac {1}{2}}\sum _{\ell \in \Omega }(x_{i}^{(\ell )}-x_{i}^{(k)})f_{j}^{(k\ell )}\right)}$

where

• ${\displaystyle k}$ and ${\displaystyle \ell }$ are atoms in the domain,
• ${\displaystyle \Omega }$ is the volume of the domain,
• ${\displaystyle m^{(k)}}$ is the mass of atom k,
• ${\displaystyle u_{i}^{(k)}}$ is the i^th component of the velocity of atom k,
• ${\displaystyle {\bar {u}}_{j}}$ is the j^th component of the average velocity of atoms in the volume,
• ${\displaystyle x_{i}^{(k)}}$ is the i^th component of the position of atom k, and
• ${\displaystyle f_{i}^{(k\ell )}}$ is the i^th component of the force applied on atom ${\displaystyle k}$ by atom ${\displaystyle \ell }$.

At zero kelvin, all velocities are zero so we have

${\displaystyle \tau _{ij}={\frac {1}{2\Omega }}\sum _{k,\ell \in \Omega }(x_{i}^{(\ell )}-x_{i}^{(k)})f_{j}^{(k\ell )}}$.

This can be thought of as follows. The τ₁₁ component of stress is the force in the x₁-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:

${\displaystyle {\mathcal {P}}_{at}=-{\frac {1}{3}}Tr(\tau ).}$

The pressure then is the ensemble average of the instantaneous pressure[2]

${\displaystyle P_{at}=\langle {\mathcal {P}}_{at}\rangle .}$

This pressure is the average pressure in the volume ${\displaystyle \Omega }$.

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.[4] 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.

The (local) instantaneous virial stress is given by:[1]

${\displaystyle \tau _{ab}({\vec {r}})=-\sum _{i=1}^{N}\delta ({\vec {r}}-{\vec {r}}^{(i)})\left(m^{(i)}u_{a}^{(i)}u_{b}^{(i)}+{\frac {1}{2}}\sum _{j=1,j\neq i}^{N}({\vec {r}}^{(i)}-{\vec {r}}^{(j)})_{a}{\vec {f}}_{b}^{(ij)}\right),}$

The virial pressure can be measured via the formulas above or using volume rescaling trial moves.[5]

• Virial theorem

• Physical Interpretation of the volume averaged Virial Stress
• Computer Simulation of Liquids