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# 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

$p=-\left.{\frac {\partial A}{\partial V}}\right\vert _{{T,N}}=k_{B}T\left.{\frac {\partial \ln Q}{\partial V}}\right\vert _{{T,N}}$

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/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

## 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 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 forces are evaluated and readily available. For pair interactions, one has (Eq. 2 in [1]):

$p={\frac {k_{B}TN}{V}}+{\frac {1}{Vd}}\overline {\sum _{{i

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

${{\mathbf f}}_{{ij}}=-{\frac {{{\mathbf r}}_{{ij}}}{r_{{ij}}}}f(r_{{ij}}),$

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}TN}{V}}+{\frac {1}{Vd}}\overline {\sum _{{i

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):

$p^{*}={\frac {\beta p}{\rho }}={\frac {pV}{Nk_{B}T}}=1-\beta {\frac {2}{3}}\pi \rho \int _{0}^{{\infty }}\left({\frac {{{\rm {d}}}\Phi (r)}{{{\rm {d}}}r}}~r\right)~{{\rm {g}}}(r)r^{2}~{{\rm {d}}}r$

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