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...) |
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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 16:15, 12 February 2008
Ref 1 Eq. 3.37:
- 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 \mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\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 N} is the 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 H} is the Hamiltonian of the system 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 \Theta} is the temperature (to convert 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 \Theta} into the more familiar Kelvin scale one divides by 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 k_B} ). The 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 Z_{(N)}} is found from the normalization condition (Ref. 1 Eq. 3.38)
- 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 \frac{1}{\Gamma_{(N)}^{(0)}Z_{(N)}} \int_V \exp \left( - \frac{U_1,..., _N}{\Theta}\right) ~{\rm d}^3r_1 ... {\rm d}^3r_N \int_{- \infty}^{\infty} \exp \left( - \frac{K_{(N)}}{\Theta}\right) ~{\rm d}^3p_1 ... {\rm d}^3p_N =1}
which leads to (Ref. 1 Eq. 3.40)
- 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_{(N)}= \frac{1}{V^N} Q_{(N)}}
where (Ref. 1 Eq. 3.41)
- 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_{(N)} = \int_V \exp \left( - \frac{U_1,..., _N}{\Theta}\right) ~{\rm d}^3r_1 ... {\rm d}^3r_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)