Virial stress

Volume averaged Definition

The instantaneous volume averaged virial stress is given by

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

• k and \ell are atoms in the domain,
• \Omega is the volume of the domain,
• 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 applied on atom k by atom ℓ.

At zero kelvin, all velocities are zero so we have \tau_{ij} = \frac{1}{2\Omega} \sum_{k,\ell \in \Omega} \left( x_i^{(\ell)} - x_i^{(k)}\right) 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:

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

The pressure then is the ensemble average of the instantaneous pressure P_{at} =\langle \mathcal{P}_{at} \rangle. This pressure is the average pressure in the volume Ω.

Equivalent Definition

It's worth noting that some articles and textbook use a slightly different but equivalent version of the equation

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

where x_i^{(k\ell)} is the i-th component of the vector oriented from the ℓ-th atoms to the k-th calculated via the difference

x_i^{k\ell} = x_i^{(k)} - x_i^{(\ell)}

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 P=-\tfrac{\partial F(N,V,T)}{\partial V} 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:

\tau_{ab}(\vec{r})=- \sum_{i=1}^N \delta\left(\vec{r} - \vec{r}^{(i)}\right) \left[m^{(i)} u^{(i)}_a u^{(i)}b + \frac{1}{2} \sum{j=1, j \neq i}^{N} \left(\vec{r}^{(i)} - \vec{r}^{(j)}\right)_a \vec{f}^{(ij)}_b \right],

Measuring the virial pressure in molecular simulations

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

References

References

1. (May 23, 2023). "Spotlight: the Virial Theorem". Medium.
2. (1991). "Computer Simulations of Liquids".
3. (1980-02-01). "" Virial " pressure of the classical one-component plasma". Journal de Physique Lettres.
4. Loison, Claire. (2005). "Numerical Simulations of a Smectic Lamellar Phase of Amphiphilic Molecules". Cuvillier Verlag.
5. (2006-10-30). "The nature of the calculation of the pressure in molecular simulations of continuous models from volume perturbations". The Journal of Chemical Physics.