Calculation of equilibrium mean values in a system of molecules interacting with a radiation field

Method of calculating the temperature averages of collective variables in the system of $N$ many-level moleculi interacting with radiation field is suggested. Decomposition of the space of states of the moleculi over the subspaces of irreducible representations of the group $U(n)$ is used. Calculations are performed in the basis of the coherent states of the unitary group, which are the quasiclassical states of the radiators in the limit $N\to\infty$. An unexpected statistical interpretation of some group-theoretical quantities is obtained. The method proposed for calculating the averages is used for determining the conditions of the phase transition into spontaneous coherent state.