# Pressure

**Pressure** (\(p\)) 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

In thermodynamics the pressure is given by

\[p = - \left.\frac{\partial A}{\partial V} \right\vert_{T,N} = k_BT \left.\frac{\partial \ln Q}{\partial V} \right\vert_{T,N}\]

where \(A\) is the Helmholtz energy function, \(V\) is the volume, \(k_B\) is the Boltzmann constant, \(T\) is the temperature and \(Q (N,V,T)\) 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

\[{\mathbf F} = \sigma_{ij} {\mathbf A}\]

where \({\mathbf F}\) is the force, \({\mathbf A}\) is the area, and \(\sigma_{ij}\) is the stress tensor, given by

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

where where \(\ \sigma_{x}\), \(\ \sigma_{y}\), and \(\ \sigma_{z}\) are normal stresses, and \(\ \tau_{xy}\), \(\ \tau_{xz}\), \(\ \tau_{yx}\), \(\ \tau_{yz}\), \(\ \tau_{zx}\), and \(\ \tau_{zy}\) are shear stresses.

## 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]}):

\[ p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, \]

where \(p\) is the pressure, \(T\) is the temperature, \(V\) is the volume and \(k_B\) 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; \(d\) is the dimension of the system (3 in the "real" world). \( {\mathbf f}_{ij} \) is the force **on** particle \(i\) exerted **by** particle \(j\), and \({\mathbf r}_{ij}\) is the vector going **from** \(i\) **to** \(j\): \({\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i\).

This relationship is readily obtained by writing the partition function in "reduced coordinates", i.e. \(x^*=x/L\), etc, then considering a "blow-up" of the system by changing the value of \(L\). 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

\[ {\mathbf f}_{ij} = - \dfrac{\mathbf {r}_{ij}}{ r_{ij}} f(r_{ij}) , \]

where \(f(r)\) the force corresponding to the intermolecular potential \(\Phi(r)\):

\[-\partial \Phi(r)/\partial r.\]

For example, for the Lennard-Jones potential, \(f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r\). Hence, the expression reduces to \[ p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. \]

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

\[p = -\left. \frac{\partial A}{\partial V}\right\vert_T \]

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**): \[p^*=\frac{\beta p}{\rho}= \frac{pV}{Nk_BT} = 1 - \beta \frac{2}{3} \pi \rho \int_0^{\infty} \left( \frac{{\rm d}\Phi(r)} {{\rm d}r}~r \right)~{\rm g}(r)r^2~{\rm d}r\]

where \(\beta := 1/k_BT\), \(\Phi(r)\) is a *central* potential and \({\rm g}(r)\) is the pair distribution function.

## References

- ↑ 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)

**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)