Irreducible cluster expansion for the logarithm of the $s$-particle density matrix of a many-boson system

Irreducible cluster expansion for the logarithm of the $s$-particle density matrix is obtained by means of integrating the variational wave functions which take into account the pair correlations. The integration is performed with the aid of the (proposed earlier) method of integration over the collective variables of $(N - s)$ particles. The relationship between the number of particles in the Bose-condensate and the structural functions of the system is established and the estimate of the number of particles for the liquid He$^4$ is found. In the model systems, the formulas obtained are exact ones up to the second order in the nonideality parameter and are valid in the case of the strong nonideality as well. The calculation of the dependence of the ground state energy of one-dimensional Bose-gas on the nonideality parameter $\gamma$ is performed. The results of the calculation are in the agreement with the exact solution in all the range of the parameter $\gamma$.