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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/hydrosatic conditions. | ||
==Thermodynamics== | ==Thermodynamics== | ||
In thermodynamics the pressure is given by | In thermodynamics the pressure is given by | ||
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where where <math>\ \sigma_{x}</math>, <math>\ \sigma_{y}</math>, and <math>\ \sigma_{z}</math> are normal stresses, and <math>\ \tau_{xy}</math>, <math>\ \tau_{xz}</math>, <math>\ \tau_{yx}</math>, <math>\ \tau_{yz}</math>, <math>\ \tau_{zx}</math>, and <math>\ \tau_{zy}</math> are shear stresess. | where where <math>\ \sigma_{x}</math>, <math>\ \sigma_{y}</math>, and <math>\ \sigma_{z}</math> are normal stresses, and <math>\ \tau_{xy}</math>, <math>\ \tau_{xz}</math>, <math>\ \tau_{yx}</math>, <math>\ \tau_{yz}</math>, <math>\ \tau_{zx}</math>, and <math>\ \tau_{zy}</math> are shear stresess. | ||
==Virial pressure== | ==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 [[Newtons laws#Newton's second law of motion |forces]] are evaluated and readily available. For pair interactions, one has | The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to [[molecular dynamics]], since [[Newtons laws#Newton's second law of motion |forces]] are evaluated and readily available. For pair interactions, one has: | ||
:<math> p = \frac{ k_B T N}{V} | :<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math> | ||
where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math> is the [[Boltzmann constant]]. | where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math> is the [[Boltzmann constant]]. | ||
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For example, for the [[Lennard-Jones model | Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to | For example, for the [[Lennard-Jones model | Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to | ||
:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ V | :<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math> | ||
Notice that most [[Realistic models |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 sphere model | hard spheres]], are always positive. | Notice that most [[Realistic models |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 sphere model | hard spheres]], are always positive. | ||
==Pressure equation== | ==Pressure equation== | ||
For particles acting through two-body central forces alone one may use the [[Thermodynamic relations | thermodynamic relation]] | For particles acting through two-body central forces alone one may use the [[Thermodynamic relations | thermodynamic relation]] | ||
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<math>\Phi(r)</math> is a ''central'' [[Intermolecular pair potential | potential]] and <math>{\rm g}(r)</math> is the [[pair distribution function]]. | <math>\Phi(r)</math> is a ''central'' [[Intermolecular pair potential | potential]] and <math>{\rm g}(r)</math> is the [[pair distribution function]]. | ||
==See also== | ==See also== | ||
*[[Test volume method]] | *[[Test volume method]] | ||
==References== | ==References== | ||
<references/> | <references/> | ||
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*[http://dx.doi.org/10.1063/1.3626410 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)] | *[http://dx.doi.org/10.1063/1.3626410 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)] | ||
*[http://dx.doi.org/10.1063/1.3692733 Péter T. Kiss and András Baranyai "On the pressure calculation for polarizable models in computer simulation", Journal of Chemical Physics '''136''' 104109 (2012)] | *[http://dx.doi.org/10.1063/1.3692733 Péter T. Kiss and András Baranyai "On the pressure calculation for polarizable models in computer simulation", Journal of Chemical Physics '''136''' 104109 (2012)] | ||