Grand canonical ensemble: Difference between revisions
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== Partition Function == | == Partition Function == | ||
''Classical'' | ''Classical'' partition function (one-component system) in a three-dimensional space: <math> Q_{\mu VT} </math> | ||
:<math> Q_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right] V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] </math> | :<math> Q_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right] V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] </math> | ||
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*<math> \left. N \right. </math> is the number of particles | *<math> \left. N \right. </math> is the number of particles | ||
* <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (depends on the temperature) | * <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (which depends on the temperature) | ||
* <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]] | * <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]] | ||
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* <math> \left. U \right. </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model) | * <math> \left. U \right. </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model) | ||
* <math> \left( R^*\right)^{3N} </math> represent the 3N position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math> | * <math> \left( R^*\right)^{3N} </math> represent the <math>3N</math> position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math> | ||
== Free energy and Partition Function == | == Free energy and Partition Function == | ||
Revision as of 16:50, 28 February 2007
Ensemble variables
- Chemical Potential,
- Volume,
- Temperature,
Partition Function
Classical partition function (one-component system) in a three-dimensional space:
![{\displaystyle Q_{\mu VT}=\sum _{N=0}^{\infty }{\frac {\exp \left[\beta \mu N\right]V^{N}}{N!\Lambda ^{3N}}}\int d(R^{*})^{3N}\exp \left[-\beta U\left(V,(R^{*})^{3N}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d47556ad6dddaae0f31b5b97afd881e5dae0724a)
where:
- is the number of particles
- is the de Broglie thermal wavelength (which depends on the temperature)
- , with being the Boltzmann constant
- is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- represent the position coordinates of the particles (reduced with the system size): i.e.
Free energy and Partition Function
(THis subsection should be checked)
The Corresponding thermodynamic potentail for the Grand Canonical Partition function is:
- , i.e.:
