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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf r}_{ij}} is the vector going from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i} .

This relationship is readily obtained by writing the partition function in "reduced coordinates" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^*=x/L} , etc, then considering a "blow-up" of the system by changing the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij}) , }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(r)} the force corresponding to the intermolecular potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(r)} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\partial u(r)/\partial r.}

E.g., for the Lennard-Jones potential, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r} . Hence, the expression reduces to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. }