Editing Gibbs ensemble

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

:<math>\mathcal{G}_{(N)} (X_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}</math>

where <math>\Gamma_{(N)}^{(0)}</math> is a normalized constant with the dimensions
of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>.

:<math>\left. X_{(N)} \right.= \{ r_1 , ...,  r_N ; p_1 , ...,  p_N \}</math>

Normalization condition (Ref. 1 Eq. 2.3):

:<math>\frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \mathcal{G}_{(N)} {\rm d}\mathcal{N} =1</math>

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

:<math>\Gamma_{(N)}^{(0)} = V^N \mathcal{P}^{3N}</math>

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

:<math>\mathcal{P} = \sqrt{2 \pi m \Theta}</math>

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

:<math>\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)}</math>

===[[Ergodic hypothesis |Ergodic theory]]===
Ref. 1 Eq. 2.6

:<math>\langle \psi \rangle = \overline \psi</math>

===[[Entropy]]===
Ref. 1 Eq. 2.70

:<math>S_{(N)}= - \frac{k_B}{ V^N \mathcal{P}^{3N}} \int_\Gamma  \Omega_1,... _N  \mathcal{G}_1,... _N {\rm d}\Gamma_{(N)}</math>

where <math>\Omega</math> is the ''N''-particle [[thermal potential]] (Ref. 1 Eq. 2.12)

:<math>\Omega_{(N)} (X_{(N)},t)= \ln \mathcal{G}_{(N)} (X_{(N)},t)</math>

==References==
# G. A. Martynov  "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
[[category: statistical mechanics]]