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  • 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.


    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.


    The SI units for pressure are Pascals (Pa), 1 Pa being 1 N/m2, or 1 J/m3. Other frequently encountered units are bars and millibars (mbar); 1 mbar = 100 Pa = 1 hPa, 1 hectopascal. 1 bar is 105 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


    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.


    1. 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)

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