Latest revision |
Your text |
Line 1: |
Line 1: |
| | Canonical Ensemble: |
| | |
| Variables: | | Variables: |
|
| |
|
Line 5: |
Line 7: |
| * Volume, <math> V </math> | | * Volume, <math> V </math> |
|
| |
|
| * [[Temperature]], <math> T </math> | | * Temperature, <math> T </math> |
|
| |
|
| == Partition Function == | | == Partition Function == |
| The [[partition function]], <math>Q</math>,
| |
| for a system of <math>N</math> identical particles each of mass <math>m</math> is given by
| |
|
| |
| :<math>Q_{NVT}=\frac{1}{N!h^{3N}}\iint d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]</math>
| |
|
| |
|
| where <math>h</math> is [[Planck constant |Planck's constant]], <math>T</math> is the [[temperature]], <math>k_B</math> is the [[Boltzmann constant]] and <math>H(p^N, r^N)</math> is the [[Hamiltonian]]
| | ''Classical'' Partition Function (one-component system) in a three-dimensional space: <math> Q_{NVT} </math> |
| corresponding to the total energy of the system.
| |
| For a classical one-component system in a three-dimensional space, <math> Q_{NVT} </math>,
| |
| is given by:
| |
|
| |
|
| :<math> Q_{NVT} = \frac{V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] ~~~~~~~~~~ \left( \frac{V}{N\Lambda^3} \gg 1 \right) </math>
| | <math> Q_{NVT} = \frac{V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] </math> |
|
| |
|
| where: | | where: |
|
| |
|
| * <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (depends on the temperature) | | * <math> \Lambda </math> is the [[de Broglie wavelength]] |
| | |
| * <math> \beta := \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]], and ''T'' the [[temperature]].
| |
| | |
| * <math> U </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>
| |
| | |
| ==See also==
| |
| *[[Ideal gas partition function]]
| |
| ==References==
| |
| <references/>
| |
| | |
| [[Category:Statistical mechanics]]
| |