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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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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 (Eq. 2 in <ref>[http://dx.doi.org/10.1063/1.2363381 Enrique de Miguel and George Jackson "The nature of the calculation of the pressure in molecular simulations of continuous models from volume perturbations", Journal of Chemical Physics '''125''' 164109 (2006)]</ref>): | 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 (Eq. 2 in <ref>[http://dx.doi.org/10.1063/1.2363381 Enrique de Miguel and George Jackson "The nature of the calculation of the pressure in molecular simulations of continuous models from volume perturbations", Journal of Chemical Physics '''125''' 164109 (2006)]</ref>): | ||
:<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} {\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. | ||
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*[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)] | ||
*[http://dx.doi.org/10.1063/1.4755946 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)] | *[http://dx.doi.org/10.1063/1.4755946 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)] | ||