# Pressure

**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.

## Thermodynamics[edit]

In thermodynamics the pressure is given by

where is the Helmholtz energy function, is the volume, is the Boltzmann constant, is the temperature and is the canonical ensemble partition function.

## Units[edit]

The SI units for pressure are Pascals (Pa), 1 Pa being 1 N/m^{2}, or 1 J/m^{3}. Other frequently encountered units are bars and millibars (mbar); 1 mbar = 100 Pa = 1 hPa, 1 hectopascal. 1 bar is 10^{5} 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

## Stress[edit]

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.

## Virial pressure[edit]

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 ^{[1]}):

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.

## Pressure equation[edit]

For particles acting through two-body central forces alone one may use the thermodynamic relation

Using this relation, along with the Helmholtz energy function and the canonical partition function, one
arrives at the so-called
**pressure equation** (also known as the **virial equation**):

where ,
is a *central* potential and is the pair distribution function.

## See also[edit]

## References[edit]

**Related reading**

- Aidan P. Thompson, Steven J. Plimpton, and William Mattson "General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions", Journal of Chemical Physics
**131**154107 (2009) - G. C. Rossi and M. Testa "The stress tensor in thermodynamics and statistical mechanics", Journal of Chemical Physics
**132**074902 (2010) - Nikhil Chandra Admal and E. B. Tadmor "Stress and heat flux for arbitrary multibody potentials: A unified framework", Journal of Chemical Physics
**134**184106 (2011) - Takenobu Nakamura, Wataru Shinoda, and Tamio Ikeshoji "Novel numerical method for calculating the pressure tensor in spherical coordinates for molecular systems", Journal of Chemical Physics
**135**094106 (2011) - Péter T. Kiss and András Baranyai "On the pressure calculation for polarizable models in computer simulation", Journal of Chemical Physics
**136**104109 (2012) - Jerry Zhijian Yang, Xiaojie Wu, and Xiantao Li "A generalized Irving–Kirkwood formula for the calculation of stress in molecular dynamics models", Journal of Chemical Physics
**137**134104 (2012) - J. P. Wittmer, H. Xu, P. Polińska, F. Weysser, and J. Baschnagel "Communication: Pressure fluctuations in isotropic solids and fluids", Journal of Chemical Physics
**138**191101 (2013) - F. J. Martínez-Ruiz, F. J. Blas, B. Mendiboure and A. I. Moreno-Ventas Bravo "Effect of dispersive long-range corrections to the pressure tensor: The vapour-liquid interfacial properties of the Lennard-Jones system revisited", Journal of Chemical Physics
**141**184701 (2014) - Sadrul Chowdhury, Sneha Abraham, Toby Hudson and Peter Harrowell "Long range stress correlations in the inherent structures of liquids at rest", Journal of Chemical Physics
**144**124508 (2016) - Ronald E. Miller, Ellad B. Tadmor, Joshua S. Gibson, Noam Bernstein and Fabio Pavia "Molecular dynamics at constant Cauchy stress", Journal of Chemical Physics
**144**184107 (2016) - E. R. Smith, D. M. Heyes, and D. Dini "Towards the Irving-Kirkwood limit of the mechanical stress tensor", Journal of Chemical Physics
**146**224109 (2017) - Matthias Krüger, Alexandre Solon, Vincent Démery, Christian M. Rohwer, and David S. Dean "Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory", Journal of Chemical Physics
**148**084503 (2018)