Gibbs ensemble

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

\[\mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}\]

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

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

Normalization condition (Ref. 1 Eq. 2.3):

\[\frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \mathcal{G}_{(N)} {\rm d}\mathcal{N} =1\]

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),

\[\mathcal{P} = \sqrt{2 \pi m \Theta}\]

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

\[\langle \psi ({\mathbf r},t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \psi ({\mathbf X}_{(N)}) \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t) {\rm d}\Gamma_{(N)} \]

Ergodic theory

Ref. 1 Eq. 2.6

\[\langle \psi \rangle = \overline \psi\]

Entropy

Ref. 1 Eq. 2.70

\[S_{(N)}= - \frac{k_B}{ V^N \mathcal{P}^{3N}} \int_\Gamma \Omega_1,... _N \mathcal{G}_1,... _N {\rm d}\Gamma_{(N)}\]

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

\[\Omega_{(N)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)\]

References

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