Difference between revisions of "Virial pressure"

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This is a common method to obtain the [[pressure]] from a general simulation (it is best suited for [[molecular dynamics]] since forces are evaluated in this case). For central interactions, one has:
 
This is a common method to obtain the [[pressure]] from a general simulation (it is best suited for [[molecular dynamics]] since forces are evaluated in this case). For central interactions, one has:
  
:<math> p  =  \frac{ k_B T  N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij}  {\mathbf r}_{ij} }, </math>
+
:<math> p  =  \frac{ k_B T  N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij}  {\mathbf r}_{ij} }, </math>
  
where one can recognize an ideal term and another one due to the [[virial]]. The underline is an average, which would be a time average in molecular dynamics, or ensamble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in actual 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 another one due to the [[virial]]. The underline is an average, which would be a time average in molecular dynamics, or ensamble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in actual 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>.

Revision as of 16:35, 6 February 2008

This is a common method to obtain the pressure from a general simulation (it is best suited for molecular dynamics since forces are evaluated in this case). For central 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 another one due to the virial. The underline is an average, which would be a time average in molecular dynamics, or ensamble average in Monte Carlo; d is the dimension of the system (3 in actual 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.