Gibbs distribution

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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.


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