Gibbs distribution
Ref 1 Eq. 3.37:
\[\mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\right)\]
where \(N\) is the number of particles, \(H\) is the Hamiltonian of the system and \(\Theta\) is the temperature (to convert \(\Theta\) into the more familiar kelvin scale one divides by the Boltzmann constant \(k_B\)). The constant \(Z_{(N)}\) is found from the normalization condition (Ref. 1 Eq. 3.38)
\[\frac{1}{\Gamma_{(N)}^{(0)}Z_{(N)}} \int_V \exp \left( - \frac{U_1,..., _N}{\Theta}\right) ~{\rm d}^3r_1 ... {\rm d}^3r_N \int_{- \infty}^{\infty} \exp \left( - \frac{K_{(N)}}{\Theta}\right) ~{\rm d}^3p_1 ... {\rm d}^3p_N =1\]
which leads to (Ref. 1 Eq. 3.40)
\[Z_{(N)}= \frac{1}{V^N} Q_{(N)}\]
where (Ref. 1 Eq. 3.41)
\[Q_{(N)} = \int_V \exp \left( - \frac{U_1,..., _N}{\Theta}\right) ~{\rm d}^3r_1 ... {\rm d}^3r_N\]
this is the statistical integral
\[Z \equiv \sum_n e^{-E_n/T}= {\rm tr} ~ e^{-H|T}\]
where \(H\) is the Hamiltonian of the system.
[edit] References
- G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
