Gibbs ensemble: Difference between revisions
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Here we have the ''N-particle distribution function'' | |||
(Ref. 1 Eq. 2.2) | |||
:<math>\mathcal{G}_{(N)} (X_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}</math> | |||
where <math>\Gamma_{(N)}^{(0)}</math> is a normalized constant with the dimensions | |||
of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>. | |||
:<math>\left. X_{(N)} \right.= \{ r_1 , ..., r_N ; p_1 , ..., p_N \}</math> | |||
Normalization condition (Ref. 1 Eq. 2.3): | |||
:<math>\frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \mathcal{G}_{(N)} {\rm d}\mathcal{N} =1</math> | |||
it is convenient to set (Ref. 1 Eq. 2.4) | |||
:<math>\Gamma_{(N)}^{(0)} = V^N \mathcal{P}^{3N}</math> | |||
where <math>V</math> is the volume of the system and <math>\mathcal{P}</math> is the characteristic momentum | |||
of the particles (Ref. 1 Eq. 3.26), | |||
:<math>\mathcal{P} = \sqrt{2 \pi m \Theta}</math> | |||
Macroscopic mean values are given by (Ref. 1 Eq. 2.5) | |||
:<math>\langle \psi (r,t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}} | |||
\int_{\Gamma_{(N)}} \psi (X_{(N)}) \mathcal{G}_{(N)} (X_{(N)},t) {\rm d}\Gamma_{(N)}</math> | |||
===Ergodic theory=== | |||
Ref. 1 Eq. 2.6 | |||
:<math>\langle \psi \rangle = \overline \psi</math> | |||
===Entropy=== | |||
Ref. 1 Eq. 2.70 | |||
:<math>S_{(N)}= - \frac{k_B}{ V^N \mathcal{P}^{3N}} \int_\Gamma \Omega_1,... _N \mathcal{G}_1,... _N {\rm d}\Gamma_{(N)}</math> | |||
where <math>\Omega</math> is the ''N''-particle [[thermal potential]] (Ref. 1 Eq. 2.12) | |||
:<math>\Omega_{(N)} (X_{(N)},t)= \ln \mathcal{G}_{(N)} (X_{(N)},t)</math> | |||
==References== | |||
# G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print) | |||
[[category: statistical mechanics]] | |||
Revision as of 10:39, 31 May 2007
Here we have the N-particle distribution function (Ref. 1 Eq. 2.2)
- 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)} (X_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}}
where is a normalized constant with the dimensions of the phase 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 \left. \Gamma_{(N)} \right.} .
- 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. X_{(N)} \right.= \{ r_1 , ..., r_N ; p_1 , ..., p_N \}}
Normalization condition (Ref. 1 Eq. 2.3):
- 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)}} \int_{\Gamma_{(N)}} \mathcal{G}_{(N)} {\rm d}\mathcal{N} =1}
it is convenient to set (Ref. 1 Eq. 2.4)
- 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 \Gamma_{(N)}^{(0)} = V^N \mathcal{P}^{3N}}
where is the volume 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 \mathcal{P}} is the characteristic momentum of the particles (Ref. 1 Eq. 3.26),
- 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{P} = \sqrt{2 \pi m \Theta}}
Macroscopic mean values are given by (Ref. 1 Eq. 2.5)
- 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 \langle \psi (r,t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \psi (X_{(N)}) \mathcal{G}_{(N)} (X_{(N)},t) {\rm d}\Gamma_{(N)}}
Ergodic theory
Ref. 1 Eq. 2.6
- 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 \langle \psi \rangle = \overline \psi}
Entropy
Ref. 1 Eq. 2.70
- 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 S_{(N)}= - \frac{k_B}{ V^N \mathcal{P}^{3N}} \int_\Gamma \Omega_1,... _N \mathcal{G}_1,... _N {\rm d}\Gamma_{(N)}}
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 \Omega} is the N-particle thermal potential (Ref. 1 Eq. 2.12)
- 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 \Omega_{(N)} (X_{(N)},t)= \ln \mathcal{G}_{(N)} (X_{(N)},t)}
References
- G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)