Canonical ensemble: Difference between revisions
Jump to navigation
Jump to search
mNo edit summary |
mNo edit summary |
||
| Line 15: | Line 15: | ||
where: | where: | ||
* <math> \Lambda </math> is the [[de Broglie 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]] | ||
| Line 28: | Line 28: | ||
:<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math> | :<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math> | ||
[[Category:Statistical mechanics]] | |||
Revision as of 12:44, 27 February 2007
Variables:
- Number of Particles,
- Volume,
- Temperature,
Partition Function
Classical Partition Function (one-component system) in a three-dimensional space:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Q_{NVT}={\frac {V^{N}}{N!\Lambda ^{3N}}}\int d(R^{*})^{3N}\exp \left[-\beta U\left(V,(R^{*})^{3N}\right)\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)
- 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 \beta = \frac{1}{k_B T} } , 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
- 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 }
Free energy and Partition Function
The Helmholtz energy function is related to the canonical partition function via:
- 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 A\left(N,V,T \right) = - k_B T \log Q_{NVT} }