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:
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; is the dimension of the system (3 in the "real" world). is the force on particle exerted by particle , and is the vector going from to : .
This relationship is readily obtained by writing the partition function in "reduced coordinates", i.e. , etc, then considering a "blow-up" of the system by changing the value of . 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
where the force corresponding to the intermolecular potential :
For example, for the Lennard-Jones potential, . Hence, the expression reduces to
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 purely repulsive potentials, such as hard spheres, are always positive.