Question

STATISTICAL MECHANICS PROBLEM (CLASSICAL CANONICAL ENSEMBLE):

Answer 1

N electric dipoles are in a recipient (volume V) and in thermal equilibrium with a thermostat (temperature T). They are in a constant electric field \[\vec{E}=E \hat{z}\]
Question: calculate the partition function of the system \[Q_N (V,T)\]

Answer 2

I have written the hamiltonian of the system:
\[H(P,l,\theta)= \sum _{i=1} ^N {{P_i^2 \over {2M}} +{l_i^2 \over {2I}}- qLE \cos \theta_i }\]
where L is the distance between the charges of the dipole, and \[l_i , I={1 \over 12} ML^2\]
are respectively the angular momentum of the i-th dipole with regard to its centre of mass (CM) and the momentum of inertia for a dipole (considered as a thin rod). P is the momentum of the CM of the i-th dipole. M=2m is the mass of the dipole.
The partition function is, in general, defined by:
\[Q_N (V,T)=\int e^{- \beta H} d^{3N}p' d^{3N}q' \]
where \[\{ {p'_i}\}_{i=1,...,3N} \{ {q'_i}\}_{i=1,...,3N}\] are the canonical momenta and positions.
In this case how should I work out the integral? Which variables should I use to calculate the integral? Can I use only θ P and l? Or should I also include the position of the CM of the dipoles R and the angle φ of their projection on the xy plane (orientation with respect to the x axis)?