Anomalous polymer dynamics is non-Markovian: memory effects and the generalized Langevin equation formulation

Any first course on polymer physics teaches that the dynamics of a tagged monomer of a polymer is anomalously subdiffusive, i.e., the mean-square displacement of a tagged monomer increases as tα for some α < 1 until the terminal relaxation time τ of the polymer. Beyond time τ the motion of the tagged monomer becomes diffusive. Classical examples of anomalous dynamics in polymer physics are single polymeric systems, such as phantom Rouse, self-avoiding Rouse, self-avoiding Zimm, reptation, translocation through a narrow pore in a membrane, and many-polymeric systems such as polymer melts. In this pedagogical paper I report that all these instances of anomalous dynamics in polymeric systems are robustly characterized by power-law memory kernels within a unified generalized Langevin equation (GLE) scheme, and therefore are non-Markovian. The exponents of the power-law memory kernels are related to the relaxation response of the polymers to local strains, and are derived from the equilibrium statistical physics of polymers. The anomalous dynamics of a tagged monomer of a polymer in these systems is then reproduced from the power-law memory kernels of the GLE via the fluctuation-dissipation theorem (FDT). Using this GLE formulation I further show that the characteristics of the drifts caused by a (weak) applied field on these polymeric systems are also obtained from the corresponding memory kernels.