Editing Gibbs distribution

Ref 1 Eq. 3.37:

:<math>\mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\right)</math>

where <math>N</math> is the number of particles, <math>H</math> is the [[Hamiltonian]] of the system
and <math>\Theta</math> is the temperature (to convert  <math>\Theta</math> into the more familiar
[[temperature |Kelvin scale]] one divides by the [[Boltzmann constant]] <math>k_B</math>).
The constant <math>Z_{(N)}</math> is found from the normalization condition (Ref. 1 Eq. 3.38)

:<math>\frac{1}{\Gamma_{(N)}^{(0)}Z_{(N)}} \int_V \exp \left( - \frac{U_1,..., _N}{\Theta}\right) ~{\rm d}^3r_1 ... {\rm d}^3r_N \int_{-  \infty}^{\infty}  \exp \left( - \frac{K_{(N)}}{\Theta}\right)  ~{\rm d}^3p_1 ... {\rm d}^3p_N =1</math>

which leads to (Ref. 1 Eq. 3.40)

:<math>Z_{(N)}= \frac{1}{V^N} Q_{(N)}</math>

where (Ref. 1 Eq. 3.41)

:<math>Q_{(N)} = \int_V \exp \left( - \frac{U_1,..., _N}{\Theta}\right) ~{\rm d}^3r_1  ... {\rm d}^3r_N</math>

this is the [[statistical integral]]

:<math>Z \equiv \sum_n e^{-E_n/T}= {\rm tr} ~ e^{-H|T}</math>

where <math>H</math> is the [[Hamiltonian]] of the system.
==References==
# G. A. Martynov  "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
[[category: statistical mechanics]]