Gibbs ensemble

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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[edit]

Ref. 1 Eq. 2.6

\langle \psi \rangle = \overline \psi

Entropy[edit]

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[edit]

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