Gibbs ensemble: Difference between revisions

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(Ref. 1 Eq. 2.2)
(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>
:<math>\mathcal{G}_{(N)} ({\mathbf 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
where <math>\Gamma_{(N)}^{(0)}</math> is a normalized constant with the dimensions
of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>.
of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>.


:<math>\left. X_{(N)} \right.= \{ r_1 , ...,  r_N ; p_1 , ...,  p_N \}</math>
:<math>{\mathbf X}_{(N)} = \{ {\mathbf r}_1 , ...,  {\mathbf r}_N ; {\mathbf p}_1 , ...,  {\mathbf p}_N \}</math>


Normalization condition (Ref. 1 Eq. 2.3):
Normalization condition (Ref. 1 Eq. 2.3):
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Macroscopic mean values are given by (Ref. 1 Eq. 2.5)
Macroscopic mean values are given by (Ref. 1 Eq. 2.5)


:<math>\langle \psi (r,t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}}  
:<math>\langle \psi ({\mathbf 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>
  \int_{\Gamma_{(N)}}  \psi  ({\mathbf X}_{(N)}) \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t) {\rm d}\Gamma_{(N)}
</math>


===Ergodic theory===
===Ergodic theory===
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where <math>\Omega</math> is the ''N''-particle [[thermal potential]] (Ref. 1 Eq. 2.12)
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>
:<math>\Omega_{(N)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)</math>


==References==
==References==
# G. A. Martynov  "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
# G. A. Martynov  "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
[[category: statistical mechanics]]
[[category: statistical mechanics]]

Revision as of 14:57, 10 July 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)} ({\mathbf X}_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{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 \Gamma_{(N)}^{(0)}} 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 {\mathbf X}_{(N)} = \{ {\mathbf r}_1 , ..., {\mathbf r}_N ; {\mathbf p}_1 , ..., {\mathbf p}_N \}}

Normalization condition (Ref. 1 Eq. 2.3):

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 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} 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)

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)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)}

References

  1. G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
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