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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/hydrostatic 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 \\

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[edit]

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 }{ V d } \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} = - \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 T  N}{V} + \frac{ 1 }{ V d } \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[edit]

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.

See also[edit]


Related reading