Canonical ensemble: Difference between revisions
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== Partition Function == | == Partition Function == | ||
'' | The ''classical'' [[partition function]] for a 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] </math> | ||
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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) | ||
Revision as of 10:48, 25 June 2007
Variables:
- Number of Particles, 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 N }
- Volume, 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 V }
- 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 T }
Partition Function
The classical partition function for a 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] }
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, and T the temperature.
- 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 }