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

Abstract

Any first course on polymer physics teaches that the dynamics of a tagged monomer of a polymeris anomalously subdiffusive, i.e., the mean-square displacement of a tagged monomer increases astα for someα < 1 until the terminalrelaxation time τ of thepolymer. Beyond time τ the motion of the tagged monomer becomes diffusive. Classical examples of anomalousdynamics in polymer physics are single polymeric systems, such as phantom Rouse,self-avoiding Rouse, self-avoiding Zimm, reptation, translocation through a narrow pore ina membrane, and many-polymeric systems such as polymer melts. In this pedagogicalpaper I report that all these instances of anomalous dynamics in polymeric systems arerobustly characterized by power-law memory kernels within a unified generalizedLangevin equation (GLE) scheme, and therefore are non-Markovian. The exponentsof the power-law memory kernels are related to the relaxation response of thepolymers to local strains, and are derived from the equilibrium statistical physicsof polymers. The anomalous dynamics of a tagged monomer of a polymer inthese systems is then reproduced from the power-law memory kernels of the GLEvia the fluctuation-dissipation theorem (FDT). Using this GLE formulation Ifurther show that the characteristics of the drifts caused by a (weak) applied fieldon these polymeric systems are also obtained from the corresponding memorykernels.