Gibbs distribution: Difference between revisions
Carl McBride (talk | contribs) (New page: Ref 1 Eq. 3.37: :<math>\mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\right)</math> where <math>N</math> is the number of particles, <math>H</math> is the [[H...) |
Carl McBride (talk | contribs) mNo edit summary |
||
| Line 5: | Line 5: | ||
where <math>N</math> is the number of particles, <math>H</math> is the [[Hamiltonian]] of the system | where <math>N</math> is the number of particles, <math>H</math> is the [[Hamiltonian]] of the system | ||
and <math>\Theta</math> is the temperature (to convert <math>\Theta</math> into the more familiar | and <math>\Theta</math> is the temperature (to convert <math>\Theta</math> into the more familiar | ||
[[Kelvin scale]] one divides by the [[Boltzmann constant]] <math>k_B</math>). | [[temperature |Kelvin scale]] one divides by the [[Boltzmann constant]] <math>k_B</math>). | ||
The constant <math>Z_{(N)}</math> is found from the normalization condition (Ref. 1 Eq. 3.38) | The constant <math>Z_{(N)}</math> is found from the normalization condition (Ref. 1 Eq. 3.38) | ||
Revision as of 17:15, 12 February 2008
Ref 1 Eq. 3.37:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {G}}_{(N)}={\frac {1}{Z_{(N)}}}\exp \left(-{\frac {H_{(N)}}{\Theta }}\right)}
where is the number of particles, is the Hamiltonian of the system and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Theta } is the temperature (to convert Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Theta } into the more familiar Kelvin scale one divides by the Boltzmann constant ). The constant Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{(N)}} is found from the normalization condition (Ref. 1 Eq. 3.38)
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{\Gamma _{(N)}^{(0)}Z_{(N)}}}\int _{V}\exp \left(-{\frac {U_{1},...,_{N}}{\Theta }}\right)~{\rm {d}}^{3}r_{1}...{\rm {d}}^{3}r_{N}\int _{-\infty }^{\infty }\exp \left(-{\frac {K_{(N)}}{\Theta }}\right)~{\rm {d}}^{3}p_{1}...{\rm {d}}^{3}p_{N}=1}
which leads to (Ref. 1 Eq. 3.40)
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{(N)}={\frac {1}{V^{N}}}Q_{(N)}}
where (Ref. 1 Eq. 3.41)
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Q_{(N)}=\int _{V}\exp \left(-{\frac {U_{1},...,_{N}}{\Theta }}\right)~{\rm {d}}^{3}r_{1}...{\rm {d}}^{3}r_{N}}
this is the statistical integral
- 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 Z \equiv \sum_n e^{-E_n/T}= {\rm tr} ~ e^{-H|T}}
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 H} is the Hamiltonian of the system.
References
- G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)