# Difference between revisions of "Virial pressure"

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− | 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 | + | 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: |

:<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 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]]. | ||

+ | |||

+ | If the interaction is central, the force is given by | ||

+ | :<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>: | ||

+ | |||

+ | :<math>-\partial u(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 | ||

+ | :<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math> | ||

+ | |||

+ | |||

[[category: classical mechanics]] | [[category: classical mechanics]] |

## Revision as of 21:22, 6 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:

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; is the dimension of the system (3 in the "real" world). is the force **on** particle exerted **by** particle , and is the vector going **from** **to** : .

This relationship is readily obtained by writing the partition function in "reduced coordinates" , etc, then considering a "blow-up" of the system by changing the value of . 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

where the force corresponding to the intermolecular potential :

E.g., for the Lennard-Jones potential, . Hence, the expression reduces to