Canonical ensemble: Difference between revisions
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* 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> | |||
:<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 <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]] | ||
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> | |||
where: | where: | ||
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* <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (depends on the temperature) | * <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (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]], 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> U </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model) | ||
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* <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 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]] | [[Category:Statistical mechanics]] |
Latest revision as of 12:16, 31 August 2011
Variables:
- Number of Particles,
- Volume,
Partition Function[edit]
The partition function, , for a system of identical particles each of mass is given by
where is Planck's constant, is the temperature, is the Boltzmann constant and is the Hamiltonian corresponding to the total energy of the system. For a classical one-component system in a three-dimensional space, , is given by:
where:
- is the de Broglie thermal wavelength (depends on the temperature)
- , with being the Boltzmann constant, and T the temperature.
- is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- represent the 3N position coordinates of the particles (reduced with the system size): i.e.