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Equilibrium Statistics of Weakly Slip-Linked Gaussian Polymer Chains

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.