Virial pressure

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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 T  N}{V} - \frac{ 1 }{ d V } \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)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r. Hence, the expression reduces to

 p  =  \frac{ k_B T  N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij})  r_{ij} }.