Gibbs distribution

Ref 1 Eq. 3.37:

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

where $N$ is the number of particles, $H$ is the Hamiltonian of the system and $\mathrm{\Theta }$ is the temperature (to convert $\mathrm{\Theta }$ into the more familiar Kelvin scale one divides by the Boltzmann constant $k_B$). The constant $Z_{(N)}$ is found from the normalization condition (Ref. 1 Eq. 3.38)

$\frac{1}{\underset{(N)}{\overset{(0)}{\mathrm{\Gamma }}}{Z}_{(N)}}{\int }_V\exp (-\frac{U_1,...{,}_N}{\mathrm{\Theta }}){\mathrm{d}}^3{r}_1...{\mathrm{d}}^3{r}_N\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\exp (-\frac{K_{(N)}}{\mathrm{\Theta }}){\mathrm{d}}^3{p}_1...{\mathrm{d}}^3{p}_N=1$

which leads to (Ref. 1 Eq. 3.40)

$Z_{(N)}=\frac{1}{V^N}{Q}_{(N)}$

where (Ref. 1 Eq. 3.41)

$Q_{(N)}={\int }_V\exp (-\frac{U_1,...{,}_N}{\mathrm{\Theta }}){\mathrm{d}}^3{r}_1...{\mathrm{d}}^3{r}_N$

this is the statistical integral

$Z\equiv {\sum }_n{e}^{-E_n/T}=\mathrm{t}\mathrm{r}{e}^{-H|T}$

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)