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> | |||
' | 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] </math> | :<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: | ||
* <math> \Lambda </math> is the [[de Broglie wavelength]] | * <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) | ||
* <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]] | |||
Latest revision as of 13:16, 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
![{\displaystyle 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0a25dcd86f1d8d5b2f0a9d4198662e8c24463c)
where is Planck's constant, is the temperature, is the Boltzmann constant 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 H(p^N, r^N)} is the Hamiltonian corresponding to the total energy of the system. For a classical one-component system in a three-dimensional space, 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 Q_{NVT} } , is given by:
- 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 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) }
where:
- 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 \Lambda } is the de Broglie thermal wavelength (depends on the temperature)
- , with 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 k_B } being the Boltzmann constant, and T the temperature.
- 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 U } is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- 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 \left( R^*\right)^{3N} } represent the 3N position coordinates of the particles (reduced with the system size): i.e. 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 \int d (R^*)^{3N} = 1 }