Pressure () is the force per unit area applied on a surface, in a direction perpendicular to that surface, i.e. the scalar part of the stress tensor under equilibrium/hydrostatic conditions.
In thermodynamics the pressure is given by
The SI units for pressure are Pascals (Pa), 1 Pa being 1 N/m2, or 1 J/m3. Other frequently encountered units are bars and millibars (mbar); 1 mbar = 100 Pa = 1 hPa, 1 hectopascal. 1 bar is 105 Pa by definition. This is very close to the standard atmosphere (atm), approximately equal to typical air pressure at earth mean sea level: atm, standard atmosphere = 101325 Pa = 101.325 kPa = 1013.25 hPa = 1.01325 bar
The stress is given by
where is the force, is the area, and is the stress tensor, given by
where where , , and are normal stresses, and , , , , , and are shear stresess.
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 (Eq. 2 in ):
where is the pressure, is the temperature, is the volume and is the Boltzmann constant. In this equation one can recognize an ideal gas contribution, 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.
For particles acting through two-body central forces alone one may use the thermodynamic relation
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