Virial pressure

The virial pressure is commonly used to obtain the pressure from a general simulation. It is particularly well suited to molecular dynamics, since forces are evaluated and readily available. For pair interactions, one has:

$p=\frac{k_{B}TN}{V}-\frac{1}{dV}\overline{{\sum }_{i<j}{\mathbf{f}}_{ij}{\mathbf{r}}_{ij}},$

where one can recognize an ideal term, and a second term due to the virial. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in Monte Carlo; $d$ is the dimension of the system (3 in the "real" world). ${\mathbf{f}}_{ij}$ is the force on particle $i$ exerted by particle $j$, and ${\mathbf{r}}_{ij}$ is the vector going from $i$ to $j$: ${\mathbf{r}}_{ij}={\mathbf{r}}_j-{\mathbf{r}}_i$.

This relationship is readily obtained by writing the partition function in "reduced coordinates" $x^{*}=x/L$, etc, then considering a "blow-up" of the system by changing the value of $L$. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the stress tensor and the surface tension, and are also used in constant-pressure Monte Carlo.

If the interaction is central, the force is given by

${\mathbf{f}}_{ij}=-\frac{{\mathbf{r}}_{ij}}{r_{ij}}f(r_{ij}),$

where $f(r)$ the force corresponding to the intermolecular potential $U(r)$:

$-\partial u(r)/\partial r.$

E.g., for the Lennard-Jones potential, $f(r)=24ϵ(2(\sigma /r{)}^{12}-(\sigma /r{)}^6)/r$. Hence, the expression reduces to

$p=\frac{k_{B}TN}{V}+\frac{1}{dV}\overline{{\sum }_{i<j}f(r_{ij})r_{ij}}.$

Notice that most realistic potentials are attractive at long ranges, hence the first correction to the ideal pressure will be a negative contribution: the second virial coefficient. On the other hand, contributions from repulsive potentials, such as hard spheres, are always positive.