# Gibbs ensemble

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

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

where ${\displaystyle \Gamma _{(N)}^{(0)}}$ is a normalized constant with the dimensions of the phase space ${\displaystyle \left.\Gamma _{(N)}\right.}$.

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

Normalization condition (Ref. 1 Eq. 2.3):

${\displaystyle {\frac {1}{\Gamma _{(N)}^{(0)}}}\int _{\Gamma _{(N)}}{\mathcal {G}}_{(N)}{\rm {d}}{\mathcal {N}}=1}$

it is convenient to set (Ref. 1 Eq. 2.4)

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

where ${\displaystyle V}$ is the volume of the system and ${\displaystyle {\mathcal {P}}}$ is the characteristic momentum of the particles (Ref. 1 Eq. 3.26),

${\displaystyle {\mathcal {P}}={\sqrt {2\pi m\Theta }}}$

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

${\displaystyle \langle \psi ({\mathbf {r} },t)\rangle ={\frac {1}{\Gamma _{(N)}^{(0)}}}\int _{\Gamma _{(N)}}\psi ({\mathbf {X} }_{(N)}){\mathcal {G}}_{(N)}({\mathbf {X} }_{(N)},t){\rm {d}}\Gamma _{(N)}}$

### Ergodic theory

Ref. 1 Eq. 2.6

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

### Entropy

Ref. 1 Eq. 2.70

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

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

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

## References

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