Gibbs ensemble

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

${\mathcal{G}}_{(N)}({\mathbf{X}}_{(N)},t)=\frac{\underset{(N)}{\overset{(0)}{\mathrm{\Gamma }}}}{\mathcal{N}}\frac{\mathrm{d}\mathcal{N}}{\mathrm{d}{\mathrm{\Gamma }}_{(N)}}$

where $\underset{(N)}{\overset{(0)}{\mathrm{\Gamma }}}$ is a normalized constant with the dimensions of the phase space ${\mathrm{\Gamma }}_{(N)}$.

${\mathbf{X}}_{(N)}=\{{\mathbf{r}}_1,...,{\mathbf{r}}_N;{\mathbf{p}}_1,...,{\mathbf{p}}_N\}$

Normalization condition (Ref. 1 Eq. 2.3):

$\frac{1}{\underset{(N)}{\overset{(0)}{\mathrm{\Gamma }}}}{\int }_{{\mathrm{\Gamma }}_{(N)}}{\mathcal{G}}_{(N)}\mathrm{d}\mathcal{N}=1$

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

$\underset{(N)}{\overset{(0)}{\mathrm{\Gamma }}}=V^N{\mathcal{P}}^{3N}$

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

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

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

$⟨\psi (\mathbf{r},t)⟩=\frac{1}{\underset{(N)}{\overset{(0)}{\mathrm{\Gamma }}}}{\int }_{{\mathrm{\Gamma }}_{(N)}}\psi ({\mathbf{X}}_{(N)}){\mathcal{G}}_{(N)}({\mathbf{X}}_{(N)},t)\mathrm{d}{\mathrm{\Gamma }}_{(N)}$

Ergodic theory

Ref. 1 Eq. 2.6

$⟨\psi ⟩=\overline{\psi }$

Entropy

Ref. 1 Eq. 2.70

$S_{(N)}=-\frac{k_B}{V^N{\mathcal{P}}^{3N}}{\int }_{\mathrm{\Gamma }}{\mathrm{\Omega }}_1,..{.}_N{\mathcal{G}}_1,..{.}_N\mathrm{d}{\mathrm{\Gamma }}_{(N)}$

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

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

References

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