Explicit analytical form for memory kernel in the generalized Langevin equation for end-to-end vector of Rouse chains

The generalized Langevin equation (GLE) provides an attractive theoretical framework for investigating the dynamics of conformational fluctuations of polymeric systems. While the memory kernel is a central function in the GLE, explicit analytical forms for this function have been challenging to obtain, even for the simple models of polymer dynamics. Here, we achieve an explicit analytical expression for the memory kernel in the GLE for the end-to-end vector of Rouse chains in the overdamped limit. Our derivation takes advantage of the finding that the dynamics of the end-to-end vector of Rouse chains with both free ends are equivalent to those of Rouse chains with one free end and the other fixed. For the latter model, we first show that the equations of motion of the Rouse modes as well as their statistical properties can be obtained under the boundary conditions where the free end is held fixed temporarily. We then analytically solve the terms associated with intrachain interactions in the GLE. By formally comparing these terms with the GLE based on the Rouse modes, we obtain an explicit expression for the memory kernel, along with analytical forms for the potential field and the random colored noise force. Our analytical memory kernel is confirmed by numerical calculations in the Laplace space and is shown to yield asymptotic behaviors that are consistent with previous studies. Finally, we utilize our analytical result to simulate the cyclization dynamics of Rouse chains and discuss the scaling of the cyclization time with chain length.