# Gibbs distribution

Ref 1 Eq. 3.37:

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

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

${\displaystyle {\frac {1}{\Gamma _{(N)}^{(0)}Z_{(N)}}}\int _{V}\exp \left(-{\frac {U_{1},...,_{N}}{\Theta }}\right)~{\rm {d}}^{3}r_{1}...{\rm {d}}^{3}r_{N}\int _{-\infty }^{\infty }\exp \left(-{\frac {K_{(N)}}{\Theta }}\right)~{\rm {d}}^{3}p_{1}...{\rm {d}}^{3}p_{N}=1}$

which leads to (Ref. 1 Eq. 3.40)

${\displaystyle Z_{(N)}={\frac {1}{V^{N}}}Q_{(N)}}$

where (Ref. 1 Eq. 3.41)

${\displaystyle Q_{(N)}=\int _{V}\exp \left(-{\frac {U_{1},...,_{N}}{\Theta }}\right)~{\rm {d}}^{3}r_{1}...{\rm {d}}^{3}r_{N}}$

this is the statistical integral

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

where ${\displaystyle H}$ is the Hamiltonian of the system.

## References

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