Editing Pressure

'''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==
In thermodynamics the pressure is given by

:<math>p = - \left.\frac{\partial A}{\partial V} \right\vert_{T,N} = k_BT \left.\frac{\partial \ln Q}{\partial V} \right\vert_{T,N}</math>

where <math>A</math> is the [[Helmholtz energy function]], <math>V</math> is the volume, <math>k_B</math> is the 
[[Boltzmann constant]], <math>T</math> is the [[temperature]] and <math>Q (N,V,T)</math>
is the [[Canonical ensemble | canonical ensemble partition function]].

==Units==

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

:<math>{\mathbf F} = \sigma_{ij} {\mathbf A}</math>

where <math>{\mathbf F}</math> is the force, 
<math>{\mathbf A}</math> is the area, and <math>\sigma_{ij}</math> is the stress tensor, given by

:<math>\sigma_{ij} \equiv \left[{\begin{matrix}
   \sigma _x & \tau _{xy} & \tau _{xz} \\
   \tau _{yx} & \sigma _y & \tau _{yz} \\
   \tau _{zx} & \tau _{zy} & \sigma _z \\
  \end{matrix}}\right]</math>

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==
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 d } \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]].
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>.

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]].

If the interaction is central, the force is given by
:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij})  , </math>
where <math>f(r)</math> the force corresponding to the [[Intermolecular pair potential |intermolecular potential]] <math>\Phi(r)</math>:

:<math>-\partial \Phi(r)/\partial r.</math>

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 d } \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.

==Pressure equation==
For particles acting through two-body central forces alone one may use the [[Thermodynamic relations | thermodynamic relation]]

:<math>p = -\left. \frac{\partial A}{\partial V}\right\vert_T </math>

Using this relation, along with the [[Helmholtz energy function]] and the [[partition function | canonical partition function]], one
arrives at the so-called 
'''pressure equation''' (also known as the '''virial equation'''):
:<math>p^*=\frac{\beta p}{\rho}= \frac{pV}{Nk_BT} = 1 - \beta \frac{2}{3} \pi  \rho \int_0^{\infty} \left( \frac{{\rm d}\Phi(r)} {{\rm d}r}~r \right)~{\rm g}(r)r^2~{\rm d}r</math>

where <math>\beta := 1/k_BT</math>,
<math>\Phi(r)</math> is a ''central'' [[Intermolecular pair potential | potential]] and <math>{\rm g}(r)</math> is the [[pair distribution function]].
==See also==
*[[Barostats]]
*[[Test volume method]]

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1063/1.3245303 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)]
*[http://dx.doi.org/10.1063/1.3316134  G. C. Rossi and M. Testa "The stress tensor in thermodynamics and statistical mechanics", Journal of Chemical Physics '''132''' 074902 (2010)]
*[http://dx.doi.org/10.1063/1.3582905 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)]
*[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/10.1063/1.4755946  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/10.1063/1.4807305  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)]
*[http://dx.doi.org/10.1063/1.4900773  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)]
*[http://dx.doi.org/10.1063/1.4944620  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)]
*[http://dx.doi.org/10.1063/1.4948711  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)]
*[http://dx.doi.org/10.1063/1.4984834 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)]
*[https://doi.org/10.1063/1.5019424 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)]



[[category: statistical mechanics]]
[[category: classical thermodynamics]]
[[category: classical mechanics]]