# 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:

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" , 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 :

E.g., 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 repulsive potentials, such as hard spheres, are always positive.