Gibbs ensemble

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

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

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Gamma _{(N)}^{(0)}} is a normalized constant with the dimensions of the phase space Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\Gamma _{(N)}\right.} .

$\left.X_{(N)}\right.=\{r_{1},...,r_{N};p_{1},...,p_{N}\}$

Normalization condition (Ref. 1 Eq. 2.3):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)

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

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {P}}={\sqrt {2\pi m\Theta }}}

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

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

Ergodic theory

Ref. 1 Eq. 2.6

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \psi \rangle = \overline \psi}

Entropy

Ref. 1 Eq. 2.70

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} is the N-particle thermal potential (Ref. 1 Eq. 2.12)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_{(N)} (X_{(N)},t)= \ln \mathcal{G}_{(N)} (X_{(N)},t)}

References

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