  We calculate the free energy and the pressure of a weakly slip-linked
Gaussian polymer chains. We show that the equilibrium statistics of a
slip-linked system is different from one of the corresponding ideal chain
system without any constraints by slip-links. It is shown that the pressure of
a slip-linked system decreases compared with the ideal system, which implies
that slip-linked chains spontaneously form aggregated cluster like compact
structures. These are qualitatively consistent with previous theoretical
analyses or multi chain simulations. We also show that repulsive potentials
between chains, which have been phenomenologically utilized in simulations, can
cancel the artificial pressure decrease.
