From graph Laplacian to ring polymer dynamics: part I

Abstract

Abstract A graph-theoretic formulation of overdamped Langevin dynamics and constitutive equations for ring polymers is developed. The polymer molecule is represented as a bead–spring network whose connectivity is described by a graph Laplacian. Starting from overdamped stochastic equations for Hookean bead–spring systems, the modal dynamics are obtained through diagonalization of the graph Laplacian, thereby establishing a direct connection between polymer topology, relaxation spectra, and viscoelastic constitutive behavior. The trimer ring polymer is first analyzed as the simplest nontrivial closed polymer network. In the symmetric case, the internal deformation subspace is shown to be isotropic, producing degenerate internal modes and a single relaxation time despite the existence of multiple springs. When the symmetry is broken through unequal spring stiffnesses or unequal Stokes frictions, the degeneracy is removed and the constitutive behavior becomes intrinsically multimode. The analysis is then generalized to a ring polymer consisting of an arbitrary number of beads with periodic connectivity. The cyclic graph Laplacian is diagonalized exactly by discrete Fourier modes, leading to explicit modal Langevin equations and closed-form relaxation spectra. The modal configuration tensors are shown to satisfy upper-convected Maxwell-type equations, and the polymer stress is demonstrated to be a multimode upper-convected Maxwell fluid whose relaxation spectrum is determined entirely by the graph-Laplacian eigenvalues. By eliminating the modal stresses, a closed constitutive equation for the total polymer stress is derived, yielding a hierarchy of higher-order upper-convected Maxwell models whose coefficients are elementary symmetric functions of the modal relaxation times. The formulation is further extended to ring polymers with unequal spring stiffnesses and unequal Stokes frictions, resulting in a generalized eigenvalue problem involving weighted graph Laplacians. The present work establishes an explicit topology–spectrum–constitutive correspondence for bead–spring polymer systems and provides a unified framework linking spectral graph theory, stochastic polymer dynamics, and continuum viscoelastic constitutive modeling.