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 central 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 $j$: ${\mathbf {r} }_{ij}={\mathbf {r} }_{j}-{\mathbf {r} }_{i}$.