Virial pressure: Difference between revisions
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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} | :<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 15:35, 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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is the dimension of the system (3 in actual world). Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf f}_{ij} } is the force on particle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} exerted by particle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf r}_{ij}} is the vector going from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i} .