Difference between revisions of "Virial pressure"

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(A bit more info)
m (Changed U to Phi.)
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where one can recognize an ideal term, 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]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.
 
where one can recognize an ideal term, 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]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.
  
This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates" <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress | stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].
+
This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress | 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
 
If the interaction is central, the force is given by
 
:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij})  , </math>
 
:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij})  , </math>
where <math>f(r)</math> the force corresponding to the intermolecular potential <math>U(r)</math>:
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where <math>f(r)</math> the force corresponding to the [[Intermolecular pair potential |intermolecular potential]] <math>\Phi(r)</math>:
  
:<math>-\partial u(r)/\partial r.</math>
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:<math>-\partial \Phi(r)/\partial r.</math>
  
E.g., for the [[Lennard-Jones model | Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to
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For example, for the [[Lennard-Jones model | Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to
 
:<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 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 repulsive potentials, such as [[hard sphere model | hard spheres]], are always positive.
+
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.
 
[[category: statistical mechanics]]
 
[[category: statistical mechanics]]

Revision as of 10:44, 7 February 2008

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 one can recognize an ideal term, 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.