# Difference between revisions of "Pressure"

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− | '''Pressure''' (<math>p</math>) 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/ | + | '''Pressure''' (<math>p</math>) 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== | ==Thermodynamics== | ||

In thermodynamics the pressure is given by | In thermodynamics the pressure is given by |

## Revision as of 14:45, 2 January 2018

**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.

## Contents

## Thermodynamics

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

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

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

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

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

## References

**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)