Gibbs distribution

Ref 1 Eq. 3.37:

${\mathcal {G}}_{(N)}={\frac {1}{Z_{(N)}}}\exp \left(-{\frac {H_{(N)}}{\Theta }}\right)$

where $N$ is the number of particles, $H$ is the Hamiltonian of the system and $\Theta$ is the temperature (to convert $\Theta$ into the more familiar Kelvin scale one divides by the Boltzmann constant $k_{B}$). 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

$Z\equiv \sum _{n}e^{-E_{n}/T}={\rm {tr}}~e^{-H|T}$

where $H$ is the Hamiltonian of the system.

References

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