Gibbs ensemble

Here we have the N-particle distribution function (Ref. 1 Eq. 2.2)

${\mathcal {G}}_{(N)}({\mathbf {X} }_{(N)},t)={\frac {\Gamma _{(N)}^{(0)}}{\mathcal {N}}}{\frac {{\rm {d}}{\mathcal {N}}}{{\rm {d}}\Gamma _{(N)}}}$

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Gamma _{(N)}^{(0)}} is a normalized constant with the dimensions of the phase space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\Gamma _{(N)}\right.} .

${\mathbf {X} }_{(N)}=\{{\mathbf {r} }_{1},...,{\mathbf {r} }_{N};{\mathbf {p} }_{1},...,{\mathbf {p} }_{N}\}$

Normalization condition (Ref. 1 Eq. 2.3):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)

$\Gamma _{(N)}^{(0)}=V^{N}{\mathcal {P}}^{3N}$

where $V$ is the volume of the system and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {P}}} is the characteristic momentum of the particles (Ref. 1 Eq. 3.26),

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {P}}={\sqrt {2\pi m\Theta }}}

Macroscopic mean values are given by (Ref. 1 Eq. 2.5)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle \psi ({\mathbf {r} },t)\rangle ={\frac {1}{\Gamma _{(N)}^{(0)}}}\int _{\Gamma _{(N)}}\psi ({\mathbf {X} }_{(N)}){\mathcal {G}}_{(N)}({\mathbf {X} }_{(N)},t){\rm {d}}\Gamma _{(N)}}

Ergodic theory

Ref. 1 Eq. 2.6

$\langle \psi \rangle ={\overline {\psi }}$

Entropy

Ref. 1 Eq. 2.70

$S_{(N)}=-{\frac {k_{B}}{V^{N}{\mathcal {P}}^{3N}}}\int _{\Gamma }\Omega _{1},..._{N}{\mathcal {G}}_{1},..._{N}{\rm {d}}\Gamma _{(N)}$

where $\Omega$ is the N-particle thermal potential (Ref. 1 Eq. 2.12)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)