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#REDIRECT[[Pressure#Virial pressure]]
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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:
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:<math> p  =  \frac{ k_B T  N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij}  {\mathbf r}_{ij} }, </math>
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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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In this equation one can recognize an [[Equation of State: Ideal Gas |ideal gas]] contribution, and a second term due to the [[Virial theorem |virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble  average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.
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This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress | stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].
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If the interaction is central, the force is given by
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:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij})  , </math>
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where <math>f(r)</math> the force corresponding to the [[Intermolecular pair potential |intermolecular potential]] <math>\Phi(r)</math>:
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:<math>-\partial \Phi(r)/\partial r.</math>
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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
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:<math> p  =  \frac{ k_B T  N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij})  r_{ij} }. </math>
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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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==See also==
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*[[Test volume method]]
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[[category: statistical mechanics]]

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