Gibbs ensemble
Here we have the N-particle distribution function (Ref. 1 Eq. 2.2)
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {G}}_{(N)}({\mathbf {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.} .
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathbf {X} }_{(N)}=\{{\mathbf {r} }_{1},...,{\mathbf {r} }_{N};{\mathbf {p} }_{1},...,{\mathbf {p} }_{N}\}}
Normalization condition (Ref. 1 Eq. 2.3):
- 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 \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)
- 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 \Gamma_{(N)}^{(0)} = V^N \mathcal{P}^{3N}}
where is the volume of the system and 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 \mathcal{P}} is the characteristic momentum of the particles (Ref. 1 Eq. 3.26),
- 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 \mathcal{P} = \sqrt{2 \pi m \Theta}}
Macroscopic mean values are given by (Ref. 1 Eq. 2.5)
- 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 ({\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
- 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)}
References
- G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)