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Any first course on polymer physics teaches that the dynamics of a taggedmonomer of a polymer is anomalously subdiffusive, i.e., the mean-squaredisplacement of a tagged monomer increases as $t^\alpha$ for some $\alpha<1$until the terminal relaxation time $\tau$ of the polymer. Beyond time $\tau$the motion of the tagged monomer becomes diffusive. Classical examples ofanomalous dynamics in polymer physics are single polymeric systems, such asphantom Rouse, self-avoiding Rouse, self-avoiding Zimm, reptation,translocation through a narrow pore in a membrane, and many-polymeric systemssuch as polymer melts. In this pedagogical paper I report that all theseinstances of anomalous dynamics in polymeric systems are robustly characterizedby power-law memory kernels within a {\it unified} Generalized LangevinEquation (GLE) scheme, and therefore, are non-Markovian. The exponents of thepower-law memory kernels are related to the relaxation response of the polymersto local strains, and are derived from the equilibrium statistical physics ofpolymers. The anomalous dynamics of a tagged monomer of a polymer in thesesystems is then reproduced from the power-law memory kernels of the GLE via thefluctuation-dissipation theorem (FDT). Using this GLE formulation I furthershow that the characteristics of the drifts caused by a (weak) applied field onthese polymeric systems are also obtained from the corresponding memorykernels.

Anomalous Polymer Dynamics Is Non-Markovian: Memory Effects and The Generalized Langevin Equation Formulation