Difference between revisions of "Virial pressure"

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:<math> p  =  \frac{ k_B T  N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij})  r_{ij} }. </math>
 
:<math> p  =  \frac{ k_B T  N}{V} + \frac{ 1 }{ d V } \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.
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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.
 
==See also==
 
==See also==
 
*[[Test volume method]]
 
*[[Test volume method]]
 
[[category: statistical mechanics]]
 
[[category: statistical mechanics]]

Revision as of 09:57, 7 October 2010

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:

 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_Bis 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 }{ 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.

See also